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A229111
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Expansion of the g.f. of A053723 in powers of the g.f. of A121591.
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16
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1, -5, 35, -275, 2275, -19255, 163925, -1385725, 11483875, -91781375, 688658785, -4581861025, 22550427925, 8852899375, -2431720493125, 47471706909725, -699843878180125, 9141002535744625, -111232778205154375, 1288777160650004375, -14372445132730778975
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OFFSET
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1,2
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COMMENTS
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In Verrill (1999) section 2.1, t = (eta(q^5) / eta(q))^6 the g.f. of A121591 and f = eta(q^5)^5 / eta(q) the g.f. of A053723.
Apart from signs, this is one of the Apery-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017
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LINKS
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Seiichi Manyama, Table of n, a(n) for n = 1..958
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5
H. Verrill, Some Congruences related to modular forms, Max Planck Institute, 1999.
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FORMULA
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n^3 * a(n+1) = -(2*n - 1)*(11*n*(n - 1) + 5) * a(n) - 125 * (n - 1)^3 * a(n-1).
a(n*p^k) == (p^3 + kronecker(p, 5)) * a(n*p^(k-1)) - kronecker(p, 5) * p^3*a(n*p^(-2)) (mod p^k) [Verrill, 1999]
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EXAMPLE
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G.f. = x - 5*x^2 + 35*x^3 - 275*x^4 + 2275*x^5 - 19255*x^6 + 163925*x^7 + ...
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PROG
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(PARI) {a(n) = my(m = n-1); if( n<1, 0, if( n<3, [1, -5][n], -( (5*(m - 1))^3*a(n-2) + (2*m - 1)*(11*(m^2 - m) +5)*a(n-1) )/ m^3))};
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CROSSREFS
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Cf. A053723, A109064, A121591.
The "Apery-like sequences" are A000172, A002893, A002895, A005258, A005259, A005260, A006077, A081085, A093388, A125143 (apart from signs), A183204, A219692, A229111 (apart from signs), A290575, A290576.
Sequence in context: A180900 A087630 A084135 * A138233 A248053 A002294
Adjacent sequences: A229108 A229109 A229110 * A229112 A229113 A229114
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KEYWORD
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sign,changed
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AUTHOR
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Michael Somos, Sep 30 2013
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STATUS
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approved
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