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%I #29 Apr 24 2023 06:48:51
%S 1,2,5,14,44,152,575,2354,10379,48902,245240,1301984,7294589,42959282,
%T 265263185,1712168654,11528506124,80783015192,588097479635,
%U 4439382164114,34699233959759,280381494540182,2339287666524440,20125268756209664,178348602246900569
%N Exponential generating function: exp(cosh(x)+2*x-1).
%C Old definition was "Second binomial transform of expansion of exp(cosh(x))".
%C Binomial transform of A081557.
%H Vincenzo Librandi, <a href="/A081558/b081558.txt">Table of n, a(n) for n = 0..200</a>
%F E.g.f.: exp(2*x) * exp(cosh(x))/e.
%p seq(coeff(series(exp(cosh(x)+2*x-1), x, n+1)*factorial(n), x, n), n = 0 .. 30); # _G. C. Greubel_, Aug 13 2019
%t With[{nn = 30}, CoefficientList[Series[Exp[Cosh[x] + 2 x - 1], {x, 0, nn}], x] Range[0, nn]!] (* _Vincenzo Librandi_, Aug 08 2013 *)
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace( exp(cosh(x)+2*x-1) )) \\ _G. C. Greubel_, Aug 13 2019
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(Cosh(x)+2*x-1) )); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Aug 13 2019
%o (Sage) [factorial(n)*( exp(cosh(x)+2*x-1) ).series(x,n+1).list()[n] for n in (0..30)] # _G. C. Greubel_, Aug 13 2019
%Y Cf. A005046, A081557.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Mar 22 2003
%E Definition edited by _N. J. A. Sloane_, Dec 12 2021
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