A317022 Expansion of e.g.f. sec(exp(x) - 1) + tan(exp(x) - 1).
1, 1, 2, 6, 25, 132, 838, 6209, 52592, 501238, 5308295, 61839954, 785915626, 10820482467, 160436371306, 2548722840218, 43188812459297, 777586865332600, 14823480294719570, 298285781617278681, 6318170247815155180, 140520406400556170514, 3274091838364580459623
Offset: 0
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..445
- N. J. A. Sloane, Transforms
- Eric Weisstein's World of Mathematics, Stirling Transform
Programs
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Maple
b:= proc(u, o) option remember; `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u)) end: a:= n-> add(b(j, 0)*Stirling2(n, j), j=0..n): seq(a(n), n=0..25); # Alois P. Heinz, Jul 19 2018 -
Mathematica
nmax = 22; CoefficientList[Series[Sec[Exp[x] - 1] + Tan[Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]! 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}] -
Python
from itertools import accumulate from sympy.functions.combinatorial.numbers import stirling def A317022(n): # generator of terms if n == 0: return 1 blist, c = (0,1), 0 for k in range(1,n+1): c += stirling(n,k)*blist[-1] blist = tuple(accumulate(reversed(blist),initial=0)) return c # Chai Wah Wu, Apr 18 2023
Formula
a(n) = Sum_{k=0..n} Stirling2(n,k)*A000111(k).
a(n) ~ n! * 4 / ((2+Pi) * (log(1+Pi/2))^(n+1)). - Vaclav Kotesovec, Sep 25 2019
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