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A270922
Coefficient of x^n in Product_{k>=1} (1 + x^k)^(k*n).
10
1, 1, 5, 28, 141, 751, 4064, 22198, 122381, 679375, 3792155, 21263331, 119679000, 675763232, 3826165838, 21715370653, 123502583565, 703694143160, 4016079632039, 22953901314649, 131366012754691, 752709483123304, 4317601694413683, 24790635783551008
OFFSET
0,3
COMMENTS
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
FORMULA
a(n) ~ c * d^n / sqrt(n), where d = 5.86811560195778704624328861800917668... and c = 0.25351514412215050116013727161633502...
a(n) = [x^n] exp(n*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 30 2018
MATHEMATICA
Table[SeriesCoefficient[Product[(1+x^k)^(k*n), {k, 1, n}], {x, 0, n}], {n, 0, 25}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Mar 26 2016
STATUS
approved