Search: a164555 -id:a164555
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0, 0, 1, 1, 7, 1, 5, 1, 13, 1, 1, 1, 901, 1, -11, 1, 3647, 1, -43825, 1, 1222387, 1, -854507, 1, 1181821001, 1, -76977925, 1, 23749461059, 1, -8615841275543, 1, 28267510484519, 1
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OFFSET
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1,5
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COMMENTS
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An autosequence is a sequence whose inverse binomial transform is the sequence signed. In integers, the oldest example is Fibonacci A000045. In fractions, A164555/A027642 is the son of 1/n via the Akiyama-Tanigawa algorithm; grandson is (A174110/A174111) = 1/2, 2/3, 1/2, 2/15, ...; see A164020. See A174341/A174342. All are from the same family.
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LINKS
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EXAMPLE
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Fractions are 0, 0, 1/6, 1/4, 7/30, 1/6, 5/42, 1/8, 13/90, 1/10, 1/66, 1/12, 901/2730, ...
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MATHEMATICA
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a[n_] := If[n <= 2, 0, Numerator[1/n - BernoulliB[n-1]]];
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PROG
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(Magma)
A181722:= func< n | n le 2 select 0 else Numerator(1/n - Bernoulli(n-1)) >;
(SageMath)
def A181722(n): return 0 if n<3 else numerator(1/n - bernoulli(n-1))
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CROSSREFS
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KEYWORD
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sign,frac
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AUTHOR
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STATUS
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approved
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0, 1, 5, 1, 31, 1, 41, 1, 31, 1, 61, 1, 3421, 1, -1, 1, 4127, 1, -43069, 1, 174941, 1, -854375, 1, 236366821, 1, -8553097, 1, 23749461899, 1, -8615841261683, 1, 7709321041727, 1, -2577687858361, 1, 26315271553055396563, 1, -2929993913841553, 1
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OFFSET
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0,3
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COMMENTS
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If n != 1, also the numerator of 1 - Bernoulli(n). The denominators are in A027642.
(There are no common factors to be canceled in the fractions.)
The numerators of 1 - Bernoulli(n) start 0, 3, 5,1, 31, ... and differ only at n=1 from this sequence.
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LINKS
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FORMULA
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a(2n+1) = 1.
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EXAMPLE
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The rationals r(n) begin: 0, 1/2, 5/6, 1, 31/30, 1, 41/42, 1, 31/30, 1, 61/66, 1, 3421/2730, 1, -1/6, 1, 4127/510, 1, -43069/798, 1, ... - Wolfdieter Lang, Aug 07 2017
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MAPLE
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A165226 := proc(n) if n = 1 then 1+bernoulli(n) ; else 1-bernoulli(n) ; end if; numer(%) ; end proc: # R. J. Mathar, Jan 16 2011
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CROSSREFS
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KEYWORD
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frac,easy,sign
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AUTHOR
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STATUS
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approved
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1, -1, 1, 0, 1, 0, 1, 0, 1, 0, 25, 0, 477481, 0, 49, 0, 13082689, 0, 1924313689, 0, 30489001321, 0, 730192467169, 0, 55867983514256281, 0, 73155570928609, 0, 564036899167989738841, 0, 74232720893311466588760025, 0, 59433630916551169012841089, 0, 6644474695172651051906689
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OFFSET
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0,11
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COMMENTS
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Squares of Bernoulli number numerators (apart from the sign flipped at a(1)).
The associated denominators of the squared Bernoulli numbers are in A172282.
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LINKS
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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1, 1, 2, 4, 6, 6, 6, 8, 90, 10, 6, 12, 210, 14, 30, 16, 30, 18, 42, 20, 770, 22, 6, 24, 13650, 26, 54, 28, 30, 30, 462, 32, 5610, 34, 210, 36, 51870, 38, 26, 40, 330, 42, 42, 44, 2070, 46, 6, 48, 324870, 50, 1122, 52, 30, 54, 43890, 56, 5510, 58, 6, 60, 930930
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OFFSET
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0,3
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COMMENTS
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The sequence A174341(n)/a(n) = 2, 1, 1/2, 1/4, 1/6, 1/6, 1/6, ... becomes 2, -1, 1/2, -1/4, 1/6,.. under inverse binomial transform: an autosequence, where each second term flips the sign.
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LINKS
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PROG
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(PARI)
B(n)=if(n!=1, bernfrac(n), -bernfrac(n));
a(n)=denominator(B(n) + 1/(n + 1));
(Python)
from sympy import bernoulli, Rational
def B(n):
return bernoulli(n) if n != 1 else -bernoulli(n)
def a(n):
return (B(n) + Rational(1, n + 1)).as_numer_denom()[1]
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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1, 1, 1, -1, 1, 1, 0, 0, -1, -1, 0, 0, 1, 1, 0, 0, -1, -1, 0, 0, 5, 5, 0, 0, -691, -691, 0, 0, 7, 7, 0, 0, -3617, -3617, 0, 0, 43867, 43867, 0, 0, -174611, -174611, 0, 0, 854513, 854513, 0, 0, -236364091, -236364091, 0, 0, 8553103, 8553103, 0, 0, -23749461029, -23749461029, 0, 0, 8615841276005
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OFFSET
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0,21
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COMMENTS
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which apart from the third term duplicates the Bernoulli numbers.
Essentially a duplication of the entries of A027641.
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LINKS
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CROSSREFS
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KEYWORD
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sign,frac,less
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A193220
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Denominators of the fourth row of Akiyama-Tanigawa algorithm leading to Bernoulli numbers A164555(n)/A027642(n).
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+20
3
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1, 30, 20, 35, 84, 84, 120, 495, 55, 286, 1092, 455, 280, 2040, 816, 969, 855, 1330, 1540, 5313, 1012, 2300, 7800, 2925, 819, 10962, 4060, 4495, 7440, 5456, 5984, 19635, 1785, 7770, 25308, 9139, 4940
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OFFSET
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0,2
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COMMENTS
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Denominators of row k=3 of the table in A051714.
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LINKS
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EXAMPLE
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The third row is 0, 1/30, 1/20, 2/35, 5/84, 5/84, 7/120, 28/495, 3/55, 15/286, 55/1092, 22/455, 13/280, ...
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MAPLE
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read("transforms3");
L := [seq(1/n, n=1..40)] ;
L1 := AKIYATANI(L) ; L2 := AKIYATANI(L1) ; L3 := AKIYATANI(L2) ;
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MATHEMATICA
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a[0, k_] := 1/(k+1); a[n_, k_] := a[n, k] = (k+1)*(a[n-1, k] - a[n-1, k+1]); Table[a[3, k], {k, 0, 36}] // Denominator (* Jean-François Alcover, Sep 18 2012 *)
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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A235774
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Let b(k) = A164555(k)/A027642(k), the sequence of "original" Bernoulli numbers with -1 instead of A164555(0)=1; then a(n) = numerator of the n-th term of the binomial transform of the b(k) sequence.
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+20
3
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-1, -1, 1, 1, 59, 3, 169, 5, 179, 7, 533, 9, 26609, 11, 79, 13, 3523, 15, 56635, 17, -168671, 19, 857273, 21, -236304031, 23, 8553247, 25, -23749438409, 27, 8615841677021, 29, -7709321025917, 31, 2577687858559, 33, -26315271552988224913
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OFFSET
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0,5
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COMMENTS
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(a(n)/A027642(n)) = -1, -1/2, 1/6, 1, 59/30, 3, 169/42, 5, 179/30, 7, 533/66, 9,.. .
Difference table for a(n)/A027642(n):
-1, -1/2, 1/6, 1, 59/30, 3, 169/42, ...
1/2, 2/3, 5/6, 29/30, 31/30, 43/42, 41/42, ... = A165161(n)/A051717(n+1)
1/6, 1/6, 2/15, 1/15, -1/105, -1/21, -1/105, ... not in the OEIS
0, -1/30, -1/15, -8/105, -4/105, 4/105, 8/105, ... etc.
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LINKS
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FORMULA
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MATHEMATICA
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b[0] = -1; b[1] = 1/2; b[n_] := BernoulliB[n]; a[n_] := Sum[Binomial[n, k]*b[k], {k, 0, n}] // Numerator; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jan 30 2014 *)
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CROSSREFS
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KEYWORD
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sign,frac
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AUTHOR
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STATUS
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approved
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A174129
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Numerators of the first column of the table of fractions generated by the Akiyama-Tanigawa transform from a first row A164555(k)/A027642(k).
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+20
2
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1, 1, -1, -1, 31, 7, -1051, -201, 56911, 18311, -24346415, -4227881, 425739604981, 2082738855, -759610463437, -1935668684041, 91825384886337407, 3104887811293639, -333936446105326262497, -8039608511660213481, 496858217433153341005061
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OFFSET
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0,5
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COMMENTS
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The first 6 rows if the table generated by iterative application of the Akiyama-Tanigawa transform starting with a header row of fractions A164555(k)/A027642(k) are:
1, 1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, ...
1/2, 2/3, 1/2, 2/15, -1/6, -1/7, 1/6, 4/15, -3/10, -25/33, 5/6, 1382/455, ...
-1/6, 1/3, 11/10, 6/5, -5/42, -13/7, -7/10, 68/15, 453/110, -175/11, ...
-1/2, -23/15, -3/10, 554/105, 365/42, -243/35, -1099/30, 548/165, 19827/110, ...
31/30, -37/15, -1171/70, -478/35, 469/6, 1247/7, -6153/22, -46708/33, ...
7/2, 599/21, -129/14, -38566/105, -20995/42, 211515/77, 524699/66, ...
The numerators of the leftmost column define the current sequence.
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LINKS
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FORMULA
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a(n) = numerator(Sum_{j=0..n} (-1)^(n-j)*j!*Stirling2(n,j)*B(j)), where B are the Bernoulli numbers A164555/A027642. - Fabián Pereyra, Jan 06 2022
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MAPLE
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read("transforms3") ;
A174129 := proc(n) Lin := [bernoulli(0), -bernoulli(1), seq(bernoulli(k), k=2..n+1)] ; for r from 1 to n do Lin := AKIYATANI(Lin) ; end do; numer(op(1, Lin)) ; end proc:
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MATHEMATICA
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a[0, k_] := a[0, k] = BernoulliB[k]; a[0, 1] = 1/2; a[n_, k_] := a[n, k] = (k+1)*(a[n-1, k] - a[n-1, k+1]); a[n_] := a[n, 0] // Numerator; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 14 2012 *)
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CROSSREFS
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KEYWORD
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frac,sign
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AUTHOR
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STATUS
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approved
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1, 1, -1, 1, 1, 1, 0, 0, -1, -1, 0, 0, 1, 1, 0, 0, -1, -1, 0, 0, 5, 5, 0, 0, -691, -691, 0, 0, 7, 7, 0, 0, -3617, -3617, 0, 0
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OFFSET
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0,21
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COMMENTS
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Essentially the same as A176144. (The signs of the third and fourth entry are swapped.)
This refers to a shuffling of the "original" Bernoulli numbers and the Bernoulli numbers in opposite order compared to the composition discussed in A176150.
The inverse binomial transform of the shuffle in A176150 was 1,0, -1/2, 0, 13/6, -20/3. The shuffling here would yield an inverse binomial transform 1, 0, -3/2, 4, -47/6, 40/3, -21, 95/3 etc.
The difference between the corresponding elements of these two binomial transforms element by element is 0, 0, 1, -4, 10, -20, 35, -56, 84, -120, 165, -220,..., a signed variant of A000292.
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LINKS
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KEYWORD
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frac,less,sign
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AUTHOR
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STATUS
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approved
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A191972
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The numerators of T(n, n+1) with T(0, m) = A164555(m)/A027642(m) and T(n, m) = T(n-1, m+1) - T(n-1, m), n >= 1, m >= 0.
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+20
1
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1, -1, 1, -4, 4, -16, 3056, -1856, 181312, -35853056, 1670556928, -39832634368, 545273832448, -19385421824, 53026545299456, -2753673793480966144, 68423881271489019904, -22654998127210332160
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OFFSET
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0,4
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COMMENTS
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For the denominators of T(n, n+1) see A190339, where detailed information can be found.
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LINKS
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FORMULA
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T(n, n+1) = T(n, n)/2.
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EXAMPLE
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T(n,n+1) = [1/2, -1/6, 1/15 , -4/105, 4/105, -16/231, 3056/15015, -1856/2145, 181312/36465, ...]
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MAPLE
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nmax:=20: mmax:=nmax: A164555:=proc(n): if n=1 then 1 else numer(bernoulli(n)) fi: end: A027642:=proc(n): if n=1 then 2 else denom(bernoulli(n)) fi: end: for m from 0 to 2*mmax do T(0, m):=A164555(m)/A027642(m) od: for n from 1 to nmax do for m from 0 to 2*mmax do T(n, m):=T(n-1, m+1)-T(n-1, m) od: od: for n from 0 to nmax do seq(T(n, m), m=0..mmax) od: seq(numer(T(n, n+1)), n=0..nmax-1); # Johannes W. Meijer, Jun 30 2011
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MATHEMATICA
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nmax = 17; b[n_] := BernoulliB[n]; b[1] = 1/2; bb = Table[b[n], {n, 0, 2*nmax+1}]; dd = Table[Differences[bb, n], {n, 1, nmax }]; a[0] = 1; a[n_] := dd[[n, n+2]] // Numerator; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Oct 02 2012 *)
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CROSSREFS
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KEYWORD
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sign,frac
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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