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A264037 Stirling transform of A077957 (aerated powers of 2) with 0 prepended [0, 1, 0, 2, 0, 4, 0, 8, ...]. 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
a(n) without the leading zero [1, 1, 3, 13, 55, ...] is the binomial transform of A264036.
LINKS
Eric Weisstein's MathWorld, Bell Polynomial.
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
EXAMPLE
G.f. = x + x^2 + 3*x^3 + 13*x^4 + 55*x^5 + 241*x^7 + 1171*x^8 + 6357*x^9 + ...
MATHEMATICA
Table[(BellB[n, Sqrt[2]] - BellB[n, -Sqrt[2]])/(2 Sqrt[2]), {n, 0, 24}]
PROG
(PARI) vector(100, n, n--; sum(k=0, n\2, 2^k*stirling(n, 2*k+1, 2))) \\ Altug Alkan, Nov 01 2015
CROSSREFS
Sequence in context: A291653 A183804 A117376 * A151318 A151212 A151213
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified May 18 15:24 EDT 2024. Contains 372664 sequences. (Running on oeis4.)