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A009153 Expansion of e.g.f. cosh(sinh(x)*exp(x)). 4
1, 0, 1, 6, 29, 140, 757, 4858, 36409, 302520, 2681769, 25018510, 245905365, 2559272196, 28264854685, 330408571202, 4065526003313, 52349977261040, 702393407898705, 9795673312888214, 141820637175889805 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
FORMULA
a(n) = Sum_{k=0..n} Stirling2(n, k)*(1+(-1)^k)/2*2^(n-k). - Vladeta Jovovic, Sep 26 2003
G.f.: 1 + Sum_{k>=0} x^(2*k+2)/Product_{i=0..2*k+2} (1-2*i*x). - Sergei N. Gladkovskii, Jan 06 2013
G.f.: 1 + x^2/( G(0)-x^2 ) where G(k) = x^2 + (4*x*k+2*x-1)*(4*x*k+4*x-1) - x^2*(4*x*k+2*x-1)*(4*x*k+4*x-1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 06 2013
a(n) ~ cosh(exp(r)*sinh(r)) * n^(n+1/2) / (r^(n+1/2) * exp(n+r) * sqrt(exp(2*r) * r * sech(exp(r)*sinh(r))^2 + (1+2*r) * tanh(exp(r)*sinh(r)))), where r is the root of the equation r*exp(r)*(cosh(r) + sinh(r))*tanh(exp(r)*sinh(r)) = n. - Vaclav Kotesovec, Aug 06 2014
MATHEMATICA
CoefficientList[Series[Cosh[Sinh[x]*E^x], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 06 2014 *)
Table[Sum[StirlingS2[n, k]*(1+(-1)^k)/2*2^(n-k), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 06 2014 after Vladeta Jovovic *)
Table[(BellB[n, 1/2] + BellB[n, -1/2]) 2^(n-1), {n, 0, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)
PROG
(PARI) a(n) = sum(k=0, n, stirling(n, k, 2)*(1+(-1)^k)/2*2^(n-k)); \\ Michel Marcus, Nov 02 2015
CROSSREFS
Sequence in context: A030221 A271753 A367469 * A012325 A125785 A186651
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Extended and signs tested by Olivier Gérard, Mar 15 1997
STATUS
approved

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Last modified June 21 04:30 EDT 2024. Contains 373540 sequences. (Running on oeis4.)