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A255672 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^(k*n).

%I A255672 #30 Jul 31 2025 03:47:43
%S A255672 1,1,7,37,215,1251,7459,44885,272727,1668313,10263057,63423482,
%T A255672 393440867,2448542136,15280435191,95588065737,599213418327,
%U A255672 3763242239317,23673166664695,149138199543613,940796936557265,5941862248557566,37568309060087582,237767215209245583
%N A255672 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^(k*n).
%C A255672 Number of partitions of n when parts i are of n*i kinds. - _Alois P. Heinz_, Nov 23 2018
%C A255672 From _Peter Bala_, Apr 18 2023: (Start)
%C A255672 The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.
%C A255672 Conjecture: the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for all primes p >= 3 and all positive integers n and k. (End)
%H A255672 Alois P. Heinz, <a href="/A255672/b255672.txt">Table of n, a(n) for n = 0..1000</a> (first 501 terms from Vaclav Kotesovec)
%F A255672 a(n) ~ c * d^n / sqrt(n), where d = 6.468409145117839606941857350154192468889057616577..., c = 0.25864792865819067933968646380369970564... . - _Vaclav Kotesovec_, Mar 01 2015
%F A255672 a(n) = [x^n] exp(n*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - _Ilya Gutkovskiy_, May 30 2018
%p A255672 b:= proc(n, k) option remember; `if`(n=0, 1, k*add(
%p A255672       b(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
%p A255672     end:
%p A255672 a:= n-> b(n$2):
%p A255672 seq(a(n), n=0..30);  # _Alois P. Heinz_, Mar 11 2015
%t A255672 Table[SeriesCoefficient[Product[1/(1-x^k)^(k*n),{k,1,n}],{x,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Mar 01 2015 *)
%Y A255672 Cf. A008485, A252782, A270913, A270922, A380290.
%Y A255672 Main diagonal of A255961.
%K A255672 nonn
%O A255672 0,3
%A A255672 _Vaclav Kotesovec_, Mar 01 2015