A270922 - OEIS

%I #14 Apr 20 2023 11:50:59

%S 1,1,5,28,141,751,4064,22198,122381,679375,3792155,21263331,119679000,

%T 675763232,3826165838,21715370653,123502583565,703694143160,

%U 4016079632039,22953901314649,131366012754691,752709483123304,4317601694413683,24790635783551008

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

%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 Vaclav Kotesovec, <a href="/A270922/b270922.txt">Table of n, a(n) for n = 0..500</a>

%F a(n) ~ c * d^n / sqrt(n), where d = 5.86811560195778704624328861800917668... and c = 0.25351514412215050116013727161633502...

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

%t Table[SeriesCoefficient[Product[(1+x^k)^(k*n), {k, 1, n}], {x, 0, n}], {n, 0, 25}]

%Y Cf. A255672, A270913, A270917, A270924.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Mar 26 2016