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A341395 Coefficient of x^(2*n) in (-1 + Product_{k>=1} (1 + x^k)^k)^n.

%I A341395 #11 Feb 20 2021 06:16:37
%S A341395 1,2,14,92,662,4872,36578,278161,2135902,16522967,128574734,
%T A341395 1005321616,7891885382,62160038813,491003317483,3888045701232,
%U A341395 30854283708670,245315312649653,1953735732991919,15583347966328833,124463844976490422,995305632560023009,7968042676400949882
%N A341395 Coefficient of x^(2*n) in (-1 + Product_{k>=1} (1 + x^k)^k)^n.
%H A341395 Alois P. Heinz, <a href="/A341395/b341395.txt">Table of n, a(n) for n = 0..1097</a>
%F A341395 a(n) ~ c * d^n / sqrt(n), where d = 8.191928734348241613884260036383361206707761707495484130816183793791732456844... and c = 0.30227512720649344220720362916140286571342247518684432176920275576011986255... - _Vaclav Kotesovec_, Feb 20 2021
%p A341395 g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(d^2/
%p A341395      `if`(d::odd, 1, 2), d=numtheory[divisors](j)), j=1..n)/n)
%p A341395     end:
%p A341395 b:= proc(n, k) option remember; `if`(k<2, g(n+1), (q->
%p A341395       add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
%p A341395     end:
%p A341395 a:= n-> b(n$2):
%p A341395 seq(a(n), n=0..22);  # _Alois P. Heinz_, Feb 10 2021
%t A341395 Join[{1}, Table[SeriesCoefficient[(-1 + Product[(1 + x^k)^k, {k, 1, 2 n}])^n, {x, 0, 2 n}], {n, 1, 22}]]
%t A341395 A[n_, k_] := A[n, k] = If[n == 0, 1, k Sum[A[n - j, k] Sum[(-1)^(j/d + 1) d^2, {d, Divisors[j]}], {j, 1, n}]/n]; T[n_, k_] := Sum[(-1)^i Binomial[k, i] A[n, k - i], {i, 0, k}]; Table[T[2 n, n], {n, 0, 22}]
%Y A341395 Cf. A026007, A257675, A270913, A270922, A324595, A341384, A341385, A341386, A341387, A341388, A341390, A341391, A341393, A341394.
%K A341395 nonn
%O A341395 0,2
%A A341395 _Ilya Gutkovskiy_, Feb 10 2021