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

0,3

Comments

Stirling transform of A000111.

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

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

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}]

Prog

(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

Crossrefs

Cf. A000111, A000667, A080832, A139134.

Sequence in context: A352428 A352436 A020112 * A143917 A009326 A001258

Adjacent sequences: A317019 A317020 A317021 * A317023 A317024 A317025

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 19 2018

Status

approved