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

0,5

Links

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

Formula

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

Maple

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

Mathematica

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}]

Prog

(Python)
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

Crossrefs

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

Sequence in context: A318494 A020088 A205772 * A074140 A128449 A037514

Adjacent sequences: A306333 A306334 A306335 * A306337 A306338 A306339

Keyword

sign

Author

Ilya Gutkovskiy, Feb 08 2019

Status

approved