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Expansion of e.g.f. exp(sec(x)*exp(x) - 1).
0

%I #6 Mar 27 2019 03:53:06

%S 1,1,3,11,53,297,1959,14499,120409,1097025,10931771,117685163,

%T 1363889133,16887554569,222672557631,3110742121059,45912214062961,

%U 713290136581697,11636755988405555,198800967493444875,3549276499518132325,66076184834921382313,1280502976522048458647

%N Expansion of e.g.f. exp(sec(x)*exp(x) - 1).

%e exp(sec(x)*exp(x) - 1) = 1 + x + 3*x^2/2! + 11*x^3/3! + 53*x^4/4! + 297*x^5/5! + 1959*x^6/6! + 14499*x^7/7! + ...

%p a:=series(exp(sec(x)*exp(x)-1),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # _Paolo P. Lava_, Mar 26 2019

%t nmax = 22; CoefficientList[Series[Exp[Sec[x] Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!

%t a[n_] := a[n] = Sum[(2 I)^k EulerE[k, 1/2 - I/2] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]

%Y Cf. A000364, A003701, A004211, A009006, A009212, A009248, A009268, A305708.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jun 08 2018