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 A092885 Number of partitions of n in which no parts are multiples of 25. 4
 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1957, 2435, 3008, 3715, 4560, 5597, 6831, 8334, 10121, 12280, 14841, 17921, 21560, 25914, 31050, 37162, 44352, 52877, 62876, 74685, 88507 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..10000 Kevin Acres, David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10. T. Horie and N. Kanou, Certain modular functions similar to the Dedekind eta function, Abh. Math. Sem. Univ. Hamburg 72 (2002), 89-117. MR1941549 (2003j:11043). Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016. FORMULA Expansion of q^(-1) * eta(q^25) / eta(q) in powers of q. Euler transform of period 25 sequence [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, ...]. Given g.f. A(x), then B(x) = x * A(x) satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^3 + v^3 - 5*(u*v)^2 - 2*u*v *(u+v) - u*v. G.f.: Product_{k>0} (1 - x^(25*k)) / (1 - x^k). a(n) ~ exp(4*Pi*sqrt(n)/5) / (5*sqrt(10)*n^(3/4)). - Vaclav Kotesovec, Oct 13 2015 a(n) = (1/n)*Sum_{k=1..n} A227131(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Jun 16 2017 EXAMPLE G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + ... G.f. = q + q^2 + 2*q^3 + 3*q^4 + 5*q^5 + 7*q^6 + 11*q^7 + 15*q^8 + 22*q^9 + 30*q^10 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 25, n, 25}] / Product[ 1 - x^k, {k, n}], {x, 0, n}]; a[ n_] := SeriesCoefficient[(QPochhammer[ x^25] / QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, May 13 2014 *) PROG (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^25 + A) / eta(x + A), n))}; (PARI) {a(n) = local(A, m); if( n<0, 0, n++; m=5; A = x + O(x^6); while( m

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Last modified January 23 06:15 EST 2019. Contains 319374 sequences. (Running on oeis4.)