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Expansion of e.g.f. sec(exp(x) - 1) + tan(exp(x) - 1).
3

%I #16 Feb 16 2025 08:33:56

%S 1,1,2,6,25,132,838,6209,52592,501238,5308295,61839954,785915626,

%T 10820482467,160436371306,2548722840218,43188812459297,

%U 777586865332600,14823480294719570,298285781617278681,6318170247815155180,140520406400556170514,3274091838364580459623

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

%C Stirling transform of A000111.

%H Alois P. Heinz, <a href="/A317022/b317022.txt">Table of n, a(n) for n = 0..445</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>

%F a(n) = Sum_{k=0..n} Stirling2(n,k)*A000111(k).

%F a(n) ~ n! * 4 / ((2+Pi) * (log(1+Pi/2))^(n+1)). - _Vaclav Kotesovec_, Sep 25 2019

%p b:= proc(u, o) option remember; `if`(u+o=0, 1,

%p add(b(o-1+j, u-j), j=1..u))

%p end:

%p a:= n-> add(b(j, 0)*Stirling2(n, j), j=0..n):

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Jul 19 2018

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

%t e[n_] := e[n] = (2 I)^n If[EvenQ[n], EulerE[n, 1/2], EulerE[n, 0] I]; a[n_] := a[n] = Sum[StirlingS2[n, k] e[k], {k, 0, n}]; Table[a[n], {n, 0, 22}]

%o (Python)

%o from itertools import accumulate

%o from sympy.functions.combinatorial.numbers import stirling

%o def A317022(n): # generator of terms

%o if n == 0: return 1

%o blist, c = (0,1), 0

%o for k in range(1,n+1):

%o c += stirling(n,k)*blist[-1]

%o blist = tuple(accumulate(reversed(blist),initial=0))

%o return c # _Chai Wah Wu_, Apr 18 2023

%Y Cf. A000111, A000667, A080832, A139134.

%K nonn,changed

%O 0,3

%A _Ilya Gutkovskiy_, Jul 19 2018