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%I #22 Aug 16 2020 16:33:09
%S 0,0,0,0,0,0,1,0,0,1,3,0,1,3,6,1,3,7,12,3,7,15,21,7,16,28,36,16,31,50,
%T 60,32,57,85,98,60,100,141,157,107,169,226,248,184,276,358,385,305,
%U 442,553,591,495,691,845,896,782,1063,1270,1343,1216,1608,1890,1993
%N Number of partitions of n into parts 4k and 4k+3 with at least one part of each type.
%H Robert Israel, <a href="/A035623/b035623.txt">Table of n, a(n) for n = 1..10000</a>
%F G.f.: (-1 + 1/Product_{k>=0} (1-x^(4k+3)))*(-1 + 1/Product_{k>=1} (1-x^(4k))). - _Robert Israel_, Feb 23 2016
%F a(n) ~ exp(Pi*sqrt(n/3)) * Pi^(3/4) / (2^(5/4) * 3^(5/8) * Gamma(1/4) * n^(9/8)). - _Vaclav Kotesovec_, May 26 2018
%p N:= 100:
%p P:= (-1 + 1/mul(1-x^(4*k+3), k=0..(N-3)/4))*(-1 + 1/mul(1-x^(4*k), k=1..N/4)):
%p S:= series(P,x,N+1):
%p seq(coeff(S,x,j),j=1..N); # _Robert Israel_, Feb 23 2016
%t nmax = 63; s1 = Range[1, nmax/4]*4; s2 = Range[0, nmax/4]*4 + 3;
%t Table[Count[IntegerPartitions[n, All, s1~Join~s2],
%t x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* _Robert Price_, Aug 06 2020 *)
%t nmax = 63; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(4 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(4 k)), {k, 1, nmax}]), {x, 0, nmax}], x] (* _Robert Price_, Aug 06 2020 *)
%Y Cf. A035441-A035468, A035618-A035622, A035624-A035699.
%K nonn
%O 1,11
%A _Olivier Gérard_
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