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%I #9 May 09 2021 03:33:40
%S 1,1,7,56,392,3087,21952,170471,1210104,9411920,66824632,513890832,
%T 3683707839,28086110472,201122377288,1534688027817,10978118077136,
%U 83158453503608,599161640356888,4508826988300152,32435340235930576,244366486039786096,1756858874561956865,13161303959340223232
%N Expansion of Product_{k>=1} (1 + 7^(k-1)*x^k).
%F a(n) = Sum_{k=0..A003056(n)} q(n,k) * 7^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.
%F a(n) ~ (-polylog(2, -1/7))^(1/4) * 7^n * exp(2*sqrt(-polylog(2, -1/7)*n)) / (4*sqrt(2*Pi/7)*n^(3/4)). - _Vaclav Kotesovec_, May 09 2021
%t nmax = 23; CoefficientList[Series[Product[(1 + 7^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] 7^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 23}]
%o (PARI) seq(n)={Vec(prod(k=1, n, 1 + 7^(k-1)*x^k + O(x*x^n)))} \\ _Andrew Howroyd_, May 08 2021
%Y Cf. A003056, A008289, A304961, A338676, A340103, A344062, A344063, A344064, A344065, A344067, A344068.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, May 08 2021