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%I #53 Jun 11 2020 10:04:02
%S 1,2,3,6,8,11,16,20,26,34,43,54,68,84,103,127,154,186,225,269,321,383,
%T 453,535,631,740,866,1012,1178,1368,1587,1835,2117,2440,2804,3217,
%U 3687,4215,4812,5487,6244,7096,8055,9128,10331,11681,13187,14870,16752,18846,21180
%N G.f.: (1 + x) * (1 + x^2) * (1 + x^3) * Product_{k>=1} (1 + x^k).
%C Number of partitions of n into distinct parts if there are two types of 1's, two types of 2's and two types of 3's.
%F a(n) = A036469(n) + A036469(n-3) - A036469(n-4) - A036469(n-7).
%F a(n) ~ 2*exp(Pi*sqrt(n/3)) / (3^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Jun 11 2020
%t nmax = 50; CoefficientList[Series[(1 + x) (1 + x^2) (1 + x^3) Product[(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t a[0] = 1; a[n_] := a[n] = (1/n) Sum[Sum[(-1)^(k/d + 1) If[d < 4, 2, 1] d, {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 50}]
%Y Cf. A000009, A000098, A022567, A036469, A052816, A329289.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Jun 07 2020
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