A264037 Stirling transform of A077957 (aerated powers of 2) with 0 prepended [0, 1, 0, 2, 0, 4, 0, 8, ...].

0, 1, 1, 3, 13, 55, 241, 1171, 6357, 37567, 236521, 1574331, 11068333, 82110535, 640794337, 5239439011, 44723250501, 397481121295, 3671081354137, 35176098791115, 349120380267421, 3583273413146647, 37975511840454673, 415004245048757299, 4670891190907818165

Offset: 0

Author

Vladimir Reshetnikov, Nov 01 2015

Keywords

• nonn

Comments

a(n) without the leading zero [1, 1, 3, 13, 55, ...] is the binomial transform of A264036.

Examples

			G.f. = x + x^2 + 3*x^3 + 13*x^4 + 55*x^5 + 241*x^7 + 1171*x^8 + 6357*x^9 + ...
		

Links

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

Crossrefs

Cf. A077957, A264036.

Cf. A024429, A357572, A357598.

Programs

Mathematica

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

PARI

vector(100, n, n--; sum(k=0, n\2, 2^k*stirling(n, 2*k+1, 2))) \\ Altug Alkan, Nov 01 2015

Formula

a(n) = Sum_{k=0..floor(n/2)} 2^k*Stirling2(n,2*k+1).

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

Bell_n(sqrt(2)) = A264036(n) + a(n)*sqrt(2).

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

a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A264036(k). - Seiichi Manyama, Oct 12 2022