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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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47
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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.
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See eta p. 3.
L. O'Brien, Modular forms and two new integer sequences at level 7, Massey University, 2016.
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]
a(n) = Sum_{k=0..n-1} (-1)^k * binomial(n-1,k)^3 * binomial(5*k-(n-1),3*(n-1)). - Seiichi Manyama, Sep 02 2020
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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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MATHEMATICA
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a[n_] := a[n] = Switch[n, 1, 1, 2, -5, _, (1/(n-1)^3) ((1-2(n-1)) (11(n-2) (n-1)+5) a[n-1] - 125 (n-2)^3 a[n-2])];
a /@ Range[21] (* Jean-François Alcover, Jan 13 2020 *)
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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))};
(PARI) {a(n) = sum(k=0, n-1, (-1)^k*binomial(n-1, k)^3*binomial(5*k-(n-1), 3*(n-1)))} \\ Seiichi Manyama, Sep 02 2020
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CROSSREFS
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Cf. A053723, A109064, A121591.
The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692, A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)
For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.
Sequence in context: A344840 A087630 A084135 * A138233 A322666 A248053
Adjacent sequences: A229108 A229109 A229110 * A229112 A229113 A229114
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KEYWORD
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sign
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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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