A341395 Coefficient of x^(2*n) in (-1 + Product_{k>=1} (1 + x^k)^k)^n.

1, 2, 14, 92, 662, 4872, 36578, 278161, 2135902, 16522967, 128574734, 1005321616, 7891885382, 62160038813, 491003317483, 3888045701232, 30854283708670, 245315312649653, 1953735732991919, 15583347966328833, 124463844976490422, 995305632560023009, 7968042676400949882

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1097

Formula

a(n) ~ c * d^n / sqrt(n), where d = 8.191928734348241613884260036383361206707761707495484130816183793791732456844... and c = 0.30227512720649344220720362916140286571342247518684432176920275576011986255... - Vaclav Kotesovec, Feb 20 2021

Maple

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(d^2/
     `if`(d::odd, 1, 2), d=numtheory[divisors](j)), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k<2, 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..22);  # Alois P. Heinz, Feb 10 2021

Mathematica

Join[{1}, Table[SeriesCoefficient[(-1 + Product[(1 + x^k)^k, {k, 1, 2 n}])^n, {x, 0, 2 n}], {n, 1, 22}]]
A[n_, k_] := A[n, k] = If[n == 0, 1, k Sum[A[n - j, k] Sum[(-1)^(j/d + 1) d^2, {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, 22}]

Crossrefs

Cf. A026007, A257675, A270913, A270922, A324595, A341384, A341385, A341386, A341387, A341388, A341390, A341391, A341393, A341394.

Sequence in context: A323561 A288470 A351857 * A072148 A297474 A270063

Adjacent sequences: A341392 A341393 A341394 * A341396 A341397 A341398

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 10 2021

Status

approved