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

Author

Vladimir Reshetnikov, Nov 01 2015

Keywords

• nonn

Comments

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

Examples

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..567
• Eric Weisstein's MathWorld, Bell Polynomial.

Crossrefs

Column k=2 of A357681.

Cf. A065143, A077957, A264037.

Programs

Mathematica

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

PARI

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

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