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A009153 Expansion of e.g.f. cosh(sinh(x)*exp(x)).

%I A009153 #54 Oct 07 2017 02:35:08
%S A009153 1,0,1,6,29,140,757,4858,36409,302520,2681769,25018510,245905365,
%T A009153 2559272196,28264854685,330408571202,4065526003313,52349977261040,
%U A009153 702393407898705,9795673312888214,141820637175889805
%N A009153 Expansion of e.g.f. cosh(sinh(x)*exp(x)).
%H A009153 Vincenzo Librandi, <a href="/A009153/b009153.txt">Table of n, a(n) for n = 0..100</a>
%H A009153 Vaclav Kotesovec, <a href="http://oeis.org/A216688/a216688.pdf">Asymptotic solution of the equations using the Lambert W-function</a>
%F A009153 a(n) = Sum_{k=0..n} Stirling2(n, k)*(1+(-1)^k)/2*2^(n-k). - _Vladeta Jovovic_, Sep 26 2003
%F A009153 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
%F A009153 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
%F A009153 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
%t A009153 CoefficientList[Series[Cosh[Sinh[x]*E^x], {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Aug 06 2014 *)
%t A009153 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_ *)
%t A009153 Table[(BellB[n, 1/2] + BellB[n, -1/2]) 2^(n-1), {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 01 2015 *)
%o A009153 (PARI) a(n) = sum(k=0, n, stirling(n, k, 2)*(1+(-1)^k)/2*2^(n-k)); \\ _Michel Marcus_, Nov 02 2015
%Y A009153 Cf. A009599, A065143.
%K A009153 nonn,easy
%O A009153 0,4
%A A009153 _R. H. Hardin_
%E A009153 Extended and signs tested by _Olivier Gérard_, Mar 15 1997