A092885 Number of partitions of n in which no parts are multiples of 25.
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
Offset: 0
Keywords
Examples
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 + ...
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.
Programs
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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 *) -
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
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