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A317022 Expansion of e.g.f. sec(exp(x) - 1) + tan(exp(x) - 1).

%I A317022 #16 Feb 16 2025 08:33:56
%S A317022 1,1,2,6,25,132,838,6209,52592,501238,5308295,61839954,785915626,
%T A317022 10820482467,160436371306,2548722840218,43188812459297,
%U A317022 777586865332600,14823480294719570,298285781617278681,6318170247815155180,140520406400556170514,3274091838364580459623
%N A317022 Expansion of e.g.f. sec(exp(x) - 1) + tan(exp(x) - 1).
%C A317022 Stirling transform of A000111.
%H A317022 Alois P. Heinz, <a href="/A317022/b317022.txt">Table of n, a(n) for n = 0..445</a>
%H A317022 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A317022 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>
%F A317022 a(n) = Sum_{k=0..n} Stirling2(n,k)*A000111(k).
%F A317022 a(n) ~ n! * 4 / ((2+Pi) * (log(1+Pi/2))^(n+1)). - _Vaclav Kotesovec_, Sep 25 2019
%p A317022 b:= proc(u, o) option remember; `if`(u+o=0, 1,
%p A317022       add(b(o-1+j, u-j), j=1..u))
%p A317022     end:
%p A317022 a:= n-> add(b(j, 0)*Stirling2(n, j), j=0..n):
%p A317022 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jul 19 2018
%t A317022 nmax = 22; CoefficientList[Series[Sec[Exp[x] - 1] + Tan[Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
%t A317022 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 A317022 (Python)
%o A317022 from itertools import accumulate
%o A317022 from sympy.functions.combinatorial.numbers import stirling
%o A317022 def A317022(n): # generator of terms
%o A317022     if n == 0: return 1
%o A317022     blist, c = (0,1), 0
%o A317022     for k in range(1,n+1):
%o A317022         c += stirling(n,k)*blist[-1]
%o A317022         blist = tuple(accumulate(reversed(blist),initial=0))
%o A317022     return c # _Chai Wah Wu_, Apr 18 2023
%Y A317022 Cf. A000111, A000667, A080832, A139134.
%K A317022 nonn
%O A317022 0,3
%A A317022 _Ilya Gutkovskiy_, Jul 19 2018