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A291693 Expansion of Product_{k>=1} (1 + x^q(k)), where q(k) = [x^k] Product_{k>=1} (1 + x^k).

%I A291693 #8 Feb 16 2025 08:33:51
%S A291693 1,2,3,5,6,8,11,13,16,19,22,26,30,34,38,44,49,54,62,67,74,83,89,98,
%T A291693 107,115,124,134,145,155,168,178,189,206,217,231,247,259,277,294,310,
%U A291693 327,345,365,382,404,424,444,470,489,513,539,561,588,613,641,670,699,729,756,791,824
%N A291693 Expansion of Product_{k>=1} (1 + x^q(k)), where q(k) = [x^k] Product_{k>=1} (1 + x^k).
%C A291693 Number of partitions of n into distinct terms of A000009, where 2 different parts of 1 and 2 different parts of 2 are available (1a, 1b, 2a, 2b, 3a, 4a, 5a, 6a, ...).
%H A291693 Robert Israel, <a href="/A291693/b291693.txt">Table of n, a(n) for n = 0..10000</a>
%H A291693 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PartitionFunctionQ.html">Partition Function Q</a>
%H A291693 <a href="/index/Par#part">Index entries for related partition-counting sequences</a>
%F A291693 G.f.: Product_{k>=1} (1 + x^A000009(k)).
%e A291693 a(3) = 5 because we have [3a], [2a, 1a], [2a, 1b], [2b, 1a] and [2b, 1b].
%p A291693 N:= 20: # to get a(0) .. a(A000009(N))
%p A291693 P:= mul(1+x^k,k=1..N):
%p A291693 R:= mul(1+x^coeff(P,x,n)),n=1..N):
%p A291693 seq(coeff(R,x,n),n=0..coeff(P,x,N)); # _Robert Israel_, Sep 01 2017
%t A291693 nmax = 61; CoefficientList[Series[Product[1 + x^PartitionsQ[k], {k, 1, nmax}], {x, 0, nmax}], x]
%Y A291693 Cf. A000009, A007279, A050342, A068006, A089254, A089259, A280253, A284908.
%K A291693 nonn
%O A291693 0,2
%A A291693 _Ilya Gutkovskiy_, Aug 30 2017