A255672 - OEIS

%I #28 Nov 03 2023 11:05:04

%S 1,1,7,37,215,1251,7459,44885,272727,1668313,10263057,63423482,

%T 393440867,2448542136,15280435191,95588065737,599213418327,

%U 3763242239317,23673166664695,149138199543613,940796936557265,5941862248557566,37568309060087582,237767215209245583

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

%C Number of partitions of n when parts i are of n*i kinds. - _Alois P. Heinz_, Nov 23 2018

%C From _Peter Bala_, Apr 18 2023: (Start)

%C 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 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 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 a(n) ~ c * d^n / sqrt(n), where d = 6.468409145117839606941857350154192468889057616577..., c = 0.25864792865819067933968646380369970564... . - _Vaclav Kotesovec_, Mar 01 2015

%F a(n) = [x^n] exp(n*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - _Ilya Gutkovskiy_, May 30 2018

%p b:= proc(n, k) option remember; `if`(n=0, 1, k*add(

%p       b(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Mar 11 2015

%t Table[SeriesCoefficient[Product[1/(1-x^k)^(k*n),{k,1,n}],{x,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Mar 01 2015 *)

%Y Cf. A008485, A252782, A270913, A270922.

%Y Main diagonal of A255961.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Mar 01 2015