A317022 -id:A317022 - cp's OEIS frontend

A306336 Expansion of e.g.f. sec(log(1 + x)) + tan(log(1 + x)).

1, 1, 0, 1, -2, 10, -50, 320, -2340, 19640, -184900, 1932500, -22187200, 277576000, -3757884000, 54732418000, -853278998000, 14176686784000, -250046057846000, 4665989766386000, -91838330641200000, 1901405069222360000, -41307212202493120000, 939523370329035440000, -22327292561388519640000

Offset: 0

Ilya Gutkovskiy, Feb 08 2019

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..440

Cf. A000111, A003708, A009007, A048994, A317022.

a:=series(sec(log(1 + x)) + tan(log(1 + x)),x=0,25): seq(n!*coeff(a,x,n),n=0..24); # Paolo P. Lava, Mar 26 2019

nmax = 24; CoefficientList[Series[Sec[Log[1 + x]] + Tan[Log[1 + x]], {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[StirlingS1[n, k] e[k], {k, 0, n}]; Table[a[n], {n, 0, 24}]

from itertools import accumulate
from sympy.functions.combinatorial.numbers import stirling
def A306336(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,kind=1,signed=True)*blist[-1]
        blist = tuple(accumulate(reversed(blist),initial=0))
    return c # Chai Wah Wu, Apr 18 2023

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000111(k).

a(n) ~ -2*(-1)^n * n! * exp(3*Pi*n/2) / (exp(3*Pi/2) - 1)^(n+1). - Vaclav Kotesovec, Feb 09 2019

A335788 Expansion of e.g.f. 2sec(exp(x)-1) - 2tan(exp(x)-1) - exp(x).

Original entry on oeis.org

1, 1, 3, 11, 49, 263, 1675, 12417, 105183, 1002475, 10616589, 123679907, 1571831251, 21640964933, 320872742611, 5097445680435, 86377624918593, 1555173730665199, 29646960589439139, 596571563234557361, 12636340495630310359

Offset: 0

Geoffrey Critzer, Jun 23 2020

nonn

a(n) is the number of ways to partition {1,2,...,n} into any number of blocks, then order the blocks so that the set of least elements of the blocks is an alternating permutation.

Michael De Vlieger, Table of n, a(n) for n = 0..199

Cf. A001250, A317022.

nn = 20; a[x_] := Tan[x] + Sec[x]; b[x_] := 2 a[x] - 1 - x;
Range[0, nn]! CoefficientList[Series[b[Exp[x] - 1], {x, 0, nn}], x]
(* Second program: *)
Array[Abs[-1 + Sum[4 StirlingS2[#, k] Abs[PolyLog[-k, I]], {k, #}]] &, 21, 0] (* Michael De Vlieger, Aug 02 2021, after Jean-François Alcover at A001250 *)

a(n) = Sum_{k=1..n} Stirling2(n,k)*A001250(k).
E.g.f.: B(exp(x)-1) where B(x) = 2(tan(x) + sec(x))-1-x.
a(n) ~ 8 * n! / ((Pi+2) * log(1 + Pi/2)^(n+1)). - Vaclav Kotesovec, Jun 24 2020

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A306336 Expansion of e.g.f. sec(log(1 + x)) + tan(log(1 + x)).

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A335788 Expansion of e.g.f. 2sec(exp(x)-1) - 2tan(exp(x)-1) - exp(x).

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cp's OEIS Frontend

A306336 Expansion of e.g.f. sec(log(1 + x)) + tan(log(1 + x)).

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A335788 Expansion of e.g.f. 2*sec(exp(x)-1) - 2*tan(exp(x)-1) - exp(x).

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A335788 Expansion of e.g.f. 2sec(exp(x)-1) - 2tan(exp(x)-1) - exp(x).