A301834 - OEIS

A301834

G.f. A(x) satisfies: A(x) = 1/(1 - x*A(x)/(1 - 4*x*A(x)/(1 - 9*x*A(x)/(1 - 16*x*A(x)/(1 - ... - k^2*x*A(x)/(1 - ...)))))), a continued fraction.

1, 1, 6, 77, 1710, 59882, 3091200, 222190789, 21227659638, 2599346122814, 396581942797668, 73721984076543090, 16398099489074850108, 4299479561194904805396, 1312142733349302902243508, 461104766297721671082897333, 184846637953491751729984324518, 83842823980101547405726058204534

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OFFSET

0,3

LINKS

Table of n, a(n) for n=0..17.

FORMULA

a(n) = [x^n] (Sum_{k>=0} A000364(k)*x^k)^(n+1)/(n + 1).

a(n) ~ 2^(4*n + 3) * n^(2*n + 1/2) / (exp(2*n) * Pi^(2*n + 1/2)). - Vaclav Kotesovec, Nov 05 2021

EXAMPLE

G.f. A(x) = 1 + x + 6*x^2 + 77*x^3 + 1710*x^4 + 59882*x^5 + 3091200*x^6 + 222190789*x^7 + 21227659638*x^8 + ...

MATHEMATICA

Table[SeriesCoefficient[(1 + Sum[Abs[EulerE[2*k]]*x^k, {k, 1, n}])^(n+1)/(n+1), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 05 2021 *)