A354288 Expansion of e.g.f. (1 + x)^(2/(1 - 2 * log(1+x))).

1, 2, 10, 72, 664, 7440, 97712, 1468768, 24825184, 465516672, 9582002688, 214642099584, 5195322070656, 135064965744384, 3752151488840448, 110892824334154752, 3473236656134243328, 114893633354895538176, 4002000861023966189568, 146388324613230926979072

Offset

0,2

Links

Table of n, a(n) for n=0..19.

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A088501(k) * binomial(n-1,k-1) * a(n-k).

a(n) = Sum_{k=0..n} 2^k * A000262(k) * Stirling1(n,k).

a(n) ~ exp(-7/8 + 1/(4*(exp(1/2) - 1)) + sqrt((2*n)/(exp(1/2) - 1))*exp(1/4) - n) * n^(n - 1/4) / (2^(3/4) * (exp(1/2) - 1)^(n + 1/4)). - Vaclav Kotesovec, May 23 2022

Mathematica

With[{nn=20}, CoefficientList[Series[(1+x)^(2/(1-2Log[1+x])), {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Oct 13 2022 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((1+x)^(2/(1-2*log(1+x)))))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, sum(k=0, j, 2^k*k!*stirling(j, k, 1))*binomial(i-1, j-1)*v[i-j+1])); v;

Crossrefs

Cf. A088819, A354289.

Cf. A000262, A088501, A354286, A354290.

Sequence in context: A321446 A111554 A177384 * A182525 A321389 A292406

Adjacent sequences: A354285 A354286 A354287 * A354289 A354290 A354291

Keyword

nonn

Author

Seiichi Manyama, May 23 2022

Status

approved