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

0,3

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<n, m*=5; A = x * subst((A * (1 - 2*A + 4*A^2 - 3*A^3 + A^4 ) / (1 + 3*A + 4*A^2 + 2*A^3 + A^4) / x)^(1/5), x, x^5)); polcoeff( 1 / (1/A - A -1), n))};

Crossrefs

Cf. A000041, A096562, A227131.

Sequence in context: A242701 A364809 A330642 * A330643 A213598 A000041

Adjacent sequences: A092882 A092883 A092884 * A092886 A092887 A092888

Keyword

nonn

Author

Michael Somos, Mar 10 2004

Status

approved