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A329289 Expansion of g.f. (1 + x) * (1 + x^2) * Product_{k>=1} (1 + x^k).

%I A329289 #48 Feb 03 2025 21:28:29
%S A329289 1,2,3,5,6,8,11,14,18,23,29,36,45,55,67,82,99,119,143,170,202,240,283,
%T A329289 333,391,457,533,621,721,835,966,1114,1282,1474,1690,1935,2213,2525,
%U A329289 2877,3274,3719,4219,4781,5409,6112,6900,7778,8758,9852,11068,12422
%N A329289 Expansion of g.f. (1 + x) * (1 + x^2) * Product_{k>=1} (1 + x^k).
%C A329289 Number of partitions of n into distinct parts if there are two types of 1's and two types of 2's.
%F A329289 a(n) = A036469(n) - A036469(n-4).
%F A329289 a(n) ~ exp(Pi*sqrt(n/3)) / (3^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Jun 11 2020
%F A329289 G.f.: A(x) = 1/2 * Sum_{n >= 0} x^((n-2)*(n-3)/2) / (Product_{k = 1..n} 1 - x^k). - _Peter Bala_, Feb 03 2025
%p A329289 series(1/2 * add( x^((n-2)*(n-3)/2) / mul(1 - x^k, k = 1..n), n = 0..12), x, 51):
%p A329289 seq(coeftayl(%, x = 0, n), n = 0..50); # _Peter Bala_, Feb 03 2025
%t A329289 nmax = 50; CoefficientList[Series[(1 + x) (1 + x^2) Product[(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t A329289 a[0] = 1; a[n_] := a[n] = (1/n) Sum[Sum[(-1)^(k/d + 1) If[d < 3, 2, 1] d, {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 50}]
%Y A329289 Cf. A000009, A000097, A022567, A036469, A052816, A096914, A329384.
%K A329289 nonn,easy
%O A329289 0,2
%A A329289 _Ilya Gutkovskiy_, Jun 07 2020