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%I #40 Jun 17 2023 22:55:25
%S 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,
%T 30,34,38,41,44,49,55,60,64,70,78,85,91,99,109,119,128,138,151,164,
%U 176,190,207,225,241,259,281,304,326,349,377,408,437,467,503,542,581
%N Number of partitions of n into parts congruent to 1 mod 4.
%H Vaclav Kotesovec, <a href="/A035451/b035451.txt">Table of n, a(n) for n = 0..10000</a>
%H James Mc Laughlin, Andrew V. Sills, Peter Zimmer, <a href="https://doi.org/10.37236/36">Rogers-Ramanujan-Slater Type Identities </a>, arXiv:1901.00946 [math.NT]
%F G.f.: 1/Product_{k>=0} (1 - x^(4*k+1)). - _Vladeta Jovovic_, Nov 22 2002
%F G.f.: Sum_{n>=0} (x^n / Product_{k=1..n} (1 - x^(4*k))). - _Joerg Arndt_, Apr 07 2011
%F 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
%F 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
%F a(n) = (1/n)*Sum_{k=1..n} A050449(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 20 2017
%F 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
%p 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
%t nmax=100; CoefficientList[Series[Product[1/(1-x^(4*k+1)),{k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Feb 26 2015 *)
%t nmax = 50; kmax = nmax/4; s = Range[0, kmax]*4 + 1;
%t Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* _Robert Price_, Aug 03 2020 *)
%Y Cf. A035462, A035382, A050449.
%Y Cf. similar sequences of number of partitions of n into parts congruent to 1 mod m: A000009 (m=2), A035382 (m=3), this sequence (m=4), A109697 (m=5), A109701 (m=6), A109703 (m=7), A277090 (m=8).
%K nonn
%O 0,6
%A _Olivier Gérard_
%E Offset changed by _N. J. A. Sloane_, Apr 11 2010
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