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From expansion of Belyi function for dodecahedron.
+10
4
0, 1, 739, 349247, 135081772, 46592981880, 14921201253592, 4536057410542618, 1326832753715385794, 376757242809990931884, 104488934104327921610570, 28428461728083557062643114, 7612584440278089046630434316, 2011372004697171339782546237013
LINKS
Index entries for linear recurrences with constant coefficients, signature (684, -157434, 12527460, -77460495, 130689144, 33211924, -130689144, -77460495, -12527460, -157434, -684, -1).
FORMULA
The Belyi function is 1728*z^5*(z^10-11*z^5-1)^5/(z^20+228*z^15+494*z^10-228*z^5+1)^3.
G.f.: x*(1+11*x-x^2)^5 / (1-228*x+494*x^2+228*x^3+x^4)^3. - Colin Barker, Jan 12 2016
PROG
(PARI) concat(0, Vec(x*(1+11*x-x^2)^5/(1-228*x+494*x^2+228*x^3+x^4)^3 + O(x^20))) \\ Colin Barker, Jan 12 2016
From expansion of Belyi function for cube.
+10
4
0, 1, 46, 1347, 32220, 686661, 13579914, 254863751, 4601440184, 80635542921, 1379999420134, 23167187812555, 382770785757588, 6239740764495309, 100556187294037314, 1604514927998181135, 25381661274646261616, 398462715169752739601, 6213273419843077690782
FORMULA
The Belyi function is -108*(z^4+1)^4*z^4/(z^8-14*z^4+1)^3.
G.f.: x*(1+x)^4 / (1-14*x+x^2)^3. - Colin Barker, Jan 12 2016
MATHEMATICA
LinearRecurrence[{42, -591, 2828, -591, 42, -1}, {0, 1, 46, 1347, 32220, 686661}, 30] (* Harvey P. Dale, Jul 03 2017 *)
PROG
(PARI) concat(0, Vec(x*(1+x)^4/(1-14*x+x^2)^3 + O(x^20))) \\ Colin Barker, Jan 12 2016
EXTENSIONS
Corrected by Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001
From expansion of Belyi function for octahedron.
+10
4
1, -46, 769, -5632, 18688, -44032, 85760, -147968, 234752, -350208, 498432, -683520, 909568, -1180672, 1500928, -1874432, 2305280, -2797568, 3355392, -3982848, 4684032, -5463040, 6323968, -7270912, 8307968, -9439232, 10668800, -12000768, 13439232, -14988288
FORMULA
The Belyi function is 1/Belyi function for cube.
a(n) = 256*(-1)^n*(8*n^3-24*n^2+25*n-9)/3 for n>2.
a(n) = -4*a(n-1)-6*a(n-2)-4*a(n-3)-a(n-4) for n>6.
G.f.: (1-14*x+x^2)^3 / (1+x)^4.
(End)
MATHEMATICA
LinearRecurrence[{-4, -6, -4, -1}, {1, -46, 769, -5632, 18688, -44032, 85760}, 30] (* Harvey P. Dale, Aug 02 2024 *)
PROG
(PARI) Vec((1-14*x+x^2)^3/(1+x)^4 + O(x^30)) \\ Colin Barker, Jan 12 2016
EXTENSIONS
Corrected by Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001
Expansion of j in powers of Gamma(5)-modular function Lambda^5.
+10
2
1, 739, 196874, 22478125, 1086128125, 35307387500, 913727546875, 20389341653125, 410010534950000, 7633186177665625, 133911227595521875, 2240979684247156250, 36090410657726350000, 563019001047724506250
REFERENCES
W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc., 42 (2005), 137-162; see Eq. (5.3).
A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 24.
H. McKean and V. Moll. Elliptic Curves, Camb. Univ. Press, p. 22.
FORMULA
G.f.: (1+228x+494x^2-228x^3+x^4)^3/(x(1-11x-x^2)^5).
EXAMPLE
j = 1/x + 739 + 196874*x + 22478125*x^2 + ... where x=Lambda^5= A078905.
MAPLE
t1:=1+228*z+494*z^2-228*z^3+z^4; t2:=-t1^3/(z*(z^2+11*z-1)^5); # t2 is Duke's g.f.
PROG
(PARI) a(n)=polcoeff((1-228*(x^3-x)+494*x^2+x^4)^3/x/(1-11*x-x^2)^5+x*O(x^n), n)
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