A264036 Stirling transform of A077957 (aerated powers of 2).

1, 0, 2, 6, 18, 70, 330, 1694, 9202, 53334, 332090, 2212782, 15638370, 116365990, 907975146, 7413080510, 63212284498, 561747543414, 5190343710746, 49752410984526, 493844719701186, 5068209425457862, 53705511911500746, 586862875255860062, 6605213319604075186

Offset

0,3

Comments

a(n) is the inverse binomial transform of A264037 without the leading zero [1, 1, 3, 13, 55, ...].

Links

Seiichi Manyama, Table of n, a(n) for n = 0..567

Eric Weisstein's MathWorld, Bell Polynomial.

Formula

a(n) = Sum_{k=0..n} A077957(k)*Stirling2(n,k).

a(n) = Sum_{k=0..floor(n/2)} 2^k*Stirling2(n,2*k).

a(n) = (Bell_n(sqrt(2)) + Bell_n(-sqrt(2)))/2, where Bell_n(x) is n-th Bell polynomial.

Bell_n(sqrt(2)) = a(n) + A264037(n)*sqrt(2).

E.g.f.: cosh(sqrt(2)*(exp(x) - 1)).

a(n) = 1; a(n) = 2 * Sum_{k=0..n-1} binomial(n-1, k) * A264037(k). - Seiichi Manyama, Oct 12 2022

a(n) ~ n^n / (2 * sqrt(1 + LambertW(n/sqrt(2))) * exp(n + sqrt(2) - n/LambertW(n/sqrt(2))) * LambertW(n/sqrt(2))^n). - Vaclav Kotesovec, Feb 16 2026

Example

G.f. = 1 + 2*x^2 + 6*x^3 + 18*x^4 + 70*x^5 + 330*x^6 + 1694*x^7 + 9202*x^8 + ...

Mathematica

Table[(BellB[n, Sqrt[2]] + BellB[n, -Sqrt[2]])/2, {n, 0, 24}]

Prog

(PARI) vector(100, n, n--; sum(k=0, n\2, 2^k*stirling(n, 2*k, 2))) \\ Altug Alkan, Nov 01 2015

Crossrefs

Column k=2 of A357681.

Cf. A065143, A077957, A264037.

Sequence in context: A177470 A060181 A131281 * A261994 A177472 A379742

Adjacent sequences: A264033 A264034 A264035 * A264037 A264038 A264039

Keyword

nonn

Author

Vladimir Reshetnikov, Nov 01 2015

Status

approved