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Search: a164555 -id:a164555
Displaying 1-10 of 139 results found. page 1 2 3 4 5 6 7 8 9 10 ... 14
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A181722 Numerator of (1/n - Bernoulli number A164555(n)/A027642(n)). +20
4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
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.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..625
OEIS Wiki, Autosequence.
EXAMPLE
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, ...
MATHEMATICA
a[n_] := If[n <= 2, 0, Numerator[1/n - BernoulliB[n-1]]];
Table[a[n], {n, 1, 34}] (* Jean-François Alcover, Jun 07 2017 *)
PROG
(Magma)
A181722:= func< n | n le 2 select 0 else Numerator(1/n - Bernoulli(n-1)) >;
[A181722(n): n in [1..40]]; // G. C. Greubel, Mar 25 2024
(SageMath)
def A181722(n): return 0 if n<3 else numerator(1/n - bernoulli(n-1))
[A181722(n) for n in range(1, 41)] # G. C. Greubel, Mar 25 2024
CROSSREFS
Cf. A000045, A027642, A164020, A164555, A174110, A174111, A174341, A174342.
KEYWORD
sign,frac
AUTHOR
Paul Curtz, Nov 17 2010
STATUS
approved
A165226 Numerator of 1 - A164555(n)/A027642(n). +20
3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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.
E.g.f. for the rationals r(n) = a(n)/A027642(n) = 1 - A164555(n)/A027642(n): exp(x)*(1 - x/(exp(x) - 1)). - Wolfdieter Lang, Aug 07 2017
LINKS
Table of n, a(n) for n=0..39.
FORMULA
|a(2n)| = A162173(n+1).
a(2n+1) = 1.
EXAMPLE
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
MAPLE
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
CROSSREFS
Cf. A162173, A164555, A027642.
KEYWORD
frac,easy,sign
AUTHOR
Paul Curtz, Sep 09 2009
STATUS
approved
A172298 a(n) = A027641(n) * A164555(n). +20
3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,11
COMMENTS
Squares of Bernoulli number numerators (apart from the sign flipped at a(1)).
The associated denominators of the squared Bernoulli numbers are in A172282.
LINKS
Table of n, a(n) for n=0..34.
CROSSREFS
Cf. A027641, A164555, A172282.
KEYWORD
sign
AUTHOR
Paul Curtz, Jan 31 2010
EXTENSIONS
Edited and extended by R. J. Mathar, Feb 02 2010
STATUS
approved
A174342 Denominator of ( A164555(n)/A027642(n) + 1/(n+1) ). +20
3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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.
LINKS
Table of n, a(n) for n=0..60.
OEIS Wiki, Autosequence
PROG
(PARI)
B(n)=if(n!=1, bernfrac(n), -bernfrac(n));
a(n)=denominator(B(n) + 1/(n + 1));
for(n=0, 60, print1(a(n), ", ")) \\ Indranil Ghosh, Jun 19 2017
(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]
[a(n) for n in range(61)] # Indranil Ghosh, Jun 19 2017
CROSSREFS
Cf. A174341 (numerators).
KEYWORD
nonn,frac
AUTHOR
Paul Curtz, Mar 16 2010
STATUS
approved
A176144 a(2n) = A164555(n). a(2n+1) = A027641(n). +20
3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,21
COMMENTS
Formally, these are the numerators of a sequence of fractions defined by alternating A164555(n)/A027642(n) with A027641(n)/A027642(n),
which apart from the third term duplicates the Bernoulli numbers.
Essentially a duplication of the entries of A027641.
LINKS
Table of n, a(n) for n=0..60.
CROSSREFS
Cf. A141056, A164020, A141056.
KEYWORD
sign,frac,less
AUTHOR
Paul Curtz, Apr 10 2010
EXTENSIONS
Edited by R. J. Mathar, Jun 07 2010
STATUS
approved
A193220 Denominators of the fourth row of Akiyama-Tanigawa algorithm leading to Bernoulli numbers A164555(n)/A027642(n). +20
3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Denominators of row k=3 of the table in A051714.
LINKS
Table of n, a(n) for n=0..36.
M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
D. Merlini, R. Sprugnoli, M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.
EXAMPLE
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, ...
MAPLE
read("transforms3");
L := [seq(1/n, n=1..40)] ;
L1 := AKIYATANI(L) ; L2 := AKIYATANI(L1) ; L3 := AKIYATANI(L2) ;
apply(denom, %) ; # R. J. Mathar, Aug 20 2011
MATHEMATICA
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 *)
CROSSREFS
Cf. A194531 (numerators).
KEYWORD
nonn,frac
AUTHOR
Paul Curtz, Jul 18 2011
STATUS
approved
A235774 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. +20
3
-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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
(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.
Compare with the array in A190339.
LINKS
Table of n, a(n) for n=0..36.
FORMULA
(a(n+1) - a(n))/A027642(n) = A165161(n)/A051717(n+1).
(A164558(n) - a(n))/A027642(n) = 2's = A007395.
(a(n) - A164555(n))/A027642(n) = n - 2 = A023444(n).
MATHEMATICA
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 *)
CROSSREFS
Cf. A164558/A027642, A165161(n)/A051717(n+1).
KEYWORD
sign,frac
AUTHOR
Paul Curtz, Jan 15 2014
STATUS
approved
A174129 Numerators of the first column of the table of fractions generated by the Akiyama-Tanigawa transform from a first row A164555(k)/A027642(k). +20
2
1, 1, -1, -1, 31, 7, -1051, -201, 56911, 18311, -24346415, -4227881, 425739604981, 2082738855, -759610463437, -1935668684041, 91825384886337407, 3104887811293639, -333936446105326262497, -8039608511660213481, 496858217433153341005061 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
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.
LINKS
Table of n, a(n) for n=0..20.
D. Merlini, R. Sprugnoli and M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.
FORMULA
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
MAPLE
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:
MATHEMATICA
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 *)
CROSSREFS
Cf. A141056 (denominators), A174110, A174111 (first row).
Cf. A164555, A027642, A048993.
KEYWORD
frac,sign
AUTHOR
Paul Curtz, Mar 09 2010
STATUS
approved
A176184 a(2n) = A027641(n). a(2n+1) = A164555(n). +20
2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,21
COMMENTS
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.
LINKS
Table of n, a(n) for n=0..35.
KEYWORD
frac,less,sign
AUTHOR
Paul Curtz, Apr 11 2010
STATUS
approved
A191972 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. +20
1
1, -1, 1, -4, 4, -16, 3056, -1856, 181312, -35853056, 1670556928, -39832634368, 545273832448, -19385421824, 53026545299456, -2753673793480966144, 68423881271489019904, -22654998127210332160 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
For the denominators of T(n, n+1) see A190339, where detailed information can be found.
LINKS
Table of n, a(n) for n=0..17.
FORMULA
T(n, n+1) = T(n, n)/2.
a(n+2) = (-1)^n*A181130(n+2)/2.
EXAMPLE
T(n,n+1) = [1/2, -1/6, 1/15 , -4/105, 4/105, -16/231, 3056/15015, -1856/2145, 181312/36465, ...]
MAPLE
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
MATHEMATICA
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 *)
CROSSREFS
Cf. A191302, A191754, A181131.
KEYWORD
sign,frac
AUTHOR
Paul Curtz, Jun 20 2011
EXTENSIONS
Thanks to R. J. Mathar by Paul Curtz, Jun 20 2011
Edited by Johannes W. Meijer, Jun 30 2011
STATUS
approved
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Last modified August 26 10:14 EDT 2024. Contains 375456 sequences. (Running on oeis4.)