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URL: https://oeis.org/A341265

⇱ A341265 - OEIS

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A341265
Coefficient of x^(2*n) in (-1 + Product_{k>=1} 1 / (1 + x^k))^n.
3
1, 0, 2, 3, 10, 25, 71, 203, 562, 1650, 4667, 13673, 39427, 115440, 336639, 987628, 2898658, 8529257, 25134200, 74173606, 219207815, 648546314, 1921045953, 5695642513, 16902924883, 50203798050, 149229323544, 443895849894, 1321292939459, 3935377071154, 11728037768186
OFFSET
0,3
LINKS
FORMULA
a(n) = A341279(2n,n).
a(n) ~ c * d^n / sqrt(n), where d = 3.03044218957412050685579849718626198523346... and c = 0.2319377657497495246637662111041144... - Vaclav Kotesovec, Feb 20 2021
MAPLE
g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
[1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, g(n+1),
(q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30); # Alois P. Heinz, Feb 07 2021
MATHEMATICA
Table[SeriesCoefficient[(-1 + 1/QPochhammer[-x, x])^n, {x, 0, 2 n}], {n, 0, 30}]
A[n_, k_] := A[n, k] = If[n == 0, 1, -k Sum[A[n - j, k] Sum[Mod[d, 2] d, {d, Divisors[j]}], {j, 1, n}]/n]; T[n_, k_] := Sum[(-1)^i Binomial[k, i] A[n, k - i], {i, 0, k}]; Table[T[2 n, n], {n, 0, 30}]
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 07 2021
STATUS
approved