A035451 Number of partitions of n into parts congruent to 1 mod 4.
1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 6, 7, 7, 8, 10, 11, 12, 13, 15, 17, 18, 20, 23, 26, 28, 30, 34, 38, 41, 44, 49, 55, 60, 64, 70, 78, 85, 91, 99, 109, 119, 128, 138, 151, 164, 176, 190, 207, 225, 241, 259, 281, 304, 326, 349, 377, 408, 437, 467, 503, 542, 581
Offset: 0
Keywords
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]
Crossrefs
Programs
-
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 *)
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
Extensions
Offset changed by N. J. A. Sloane, Apr 11 2010