OFFSET
0,6
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
James Mc Laughlin, Andrew V. Sills, Peter Zimmer, Rogers-Ramanujan-Slater Type Identities , arXiv:1901.00946 [math.NT]
FORMULA
G.f.: 1/Product_{k>=0} (1 - x^(4*k+1)). - Vladeta Jovovic, Nov 22 2002
G.f.: Sum_{n>=0} (x^n / Product_{k=1..n} (1 - x^(4*k))). - Joerg Arndt, Apr 07 2011
G.f.: 1 + Sum_{n>=0} (x^(4*n+1) / Product_{k>=n} (1 - x^(4*k+1))) = 1 + Sum_{n>=0} (x^(4*n+1) / Product_{k=0..n} (1 - x^(4*k+1))). - Joerg Arndt, Apr 08 2011
a(n) ~ Gamma(1/4) * exp(Pi*sqrt(n/6)) / (2^(19/8) * 3^(1/8) * n^(5/8) * Pi^(3/4)) * (1 + (Pi/(96*sqrt(6)) - 5*sqrt(3/2)/(16*Pi)) / sqrt(n)). - Vaclav Kotesovec, Feb 26 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A050449(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017
G.f.: Sum_{n>=0} x^(n*(4*n-3))/Product_{k = 1..n} ( (1-x^(4*k))*(1-x^(4*k-3)) ). (Set z = x and q = x^4 in Mc Laughlin et al., Section 1.3, Entry 7.) - Peter Bala, Feb 02 2021
MAPLE
g := add(x^(n*(4*n-3))/mul((1-x^(4*k))*(1-x^(4*k-3)), k = 1..n), n = 0..5): gser := series(g, x, 101): seq(coeff(gser, x, n), n = 0..100); # Peter Bala, Feb 02 2021
MATHEMATICA
nmax=100; CoefficientList[Series[Product[1/(1-x^(4*k+1)), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 26 2015 *)
nmax = 50; kmax = nmax/4; s = Range[0, kmax]*4 + 1;
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 03 2020 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Offset changed by N. J. A. Sloane, Apr 11 2010
STATUS
approved