cp's OEIS Frontend

A035449 Number of partitions of n into parts 8k+1 or 8k+3.

%I A035449 #22 Aug 06 2024 22:48:47
%S A035449 1,1,2,2,2,3,3,3,5,5,6,8,8,9,11,11,13,16,17,20,23,25,28,31,34,38,43,
%T A035449 48,53,59,65,70,78,86,93,105,115,125,139,150,162,179,193,211,233,251,
%U A035449 274,298,320,348,377,407,443,480,519,561,604,651,700,755,815,876,946
%N A035449 Number of partitions of n into parts 8k+1 or 8k+3.
%H A035449 Robert Israel, <a href="/A035449/b035449.txt">Table of n, a(n) for n = 1..10000</a>
%F A035449 G.f.: Product_{k>=0} 1/((1-x^(8*k+1))*(1-x^(8*k+3))). - _Robert Israel_, Aug 29 2018
%F A035449 a(n) ~ exp(Pi*sqrt(n/6)) * Gamma(3/8) * Gamma(1/8) / (8 * Pi^(3/2) * sqrt(2*n)). - _Vaclav Kotesovec_, Aug 26 2015
%p A035449 N:= 100: # for a(1)..a(N)
%p A035449 P:= 1/mul((1-x^(8*k+1))*(1-x^(8*k+3)),k=0..floor((N-1)/8)):
%p A035449 S:= series(P,x,N+1):
%p A035449 seq(coeff(S,x,j),j=1..N); # _Robert Israel_, Aug 28 2018
%t A035449 nmax = 100; Rest[CoefficientList[Series[Product[1/((1 - x^(8k+1))*(1 - x^(8k+3))), {k, 0, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Aug 26 2015 *)
%t A035449 nmax = 60; kmax = nmax/8;
%t A035449 s = Flatten[{Range[0, kmax]*8 + 1}~Join~{Range[0, kmax]*8 + 3}];
%t A035449 Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 1, nmax}] (* _Robert Price_, Aug 03 2020 *)
%Y A035449 Cf. A035679.
%K A035449 nonn
%O A035449 1,3
%A A035449 _Olivier Gérard_