OFFSET
0,3
COMMENTS
Number of partitions of n when parts i are of n*i kinds. - Alois P. Heinz, Nov 23 2018
From Peter Bala, Apr 18 2023: (Start)
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.
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)
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 501 terms from Vaclav Kotesovec)
FORMULA
a(n) ~ c * d^n / sqrt(n), where d = 6.468409145117839606941857350154192468889057616577..., c = 0.25864792865819067933968646380369970564... . - Vaclav Kotesovec, Mar 01 2015
a(n) = [x^n] exp(n*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 30 2018
MAPLE
b:= proc(n, k) option remember; `if`(n=0, 1, k*add(
b(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
end:
a:= n-> b(n$2):
seq(a(n), n=0..30); # Alois P. Heinz, Mar 11 2015
MATHEMATICA
Table[SeriesCoefficient[Product[1/(1-x^k)^(k*n), {k, 1, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 01 2015 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Mar 01 2015
STATUS
approved