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A264036 Stirling transform of A077957 (aerated powers of 2). 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
a(n) is the inverse binomial transform of A264037 without the leading zero [1, 1, 3, 13, 55, ...].
LINKS
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
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.
Sequence in context: A177470 A060181 A131281 * A261994 A177472 A364849
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified May 25 10:09 EDT 2024. Contains 372786 sequences. (Running on oeis4.)