A306336 - OEIS

%I #14 Apr 19 2023 09:04:30

%S 1,1,0,1,-2,10,-50,320,-2340,19640,-184900,1932500,-22187200,

%T 277576000,-3757884000,54732418000,-853278998000,14176686784000,

%U -250046057846000,4665989766386000,-91838330641200000,1901405069222360000,-41307212202493120000,939523370329035440000,-22327292561388519640000

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

%H Vaclav Kotesovec, <a href="/A306336/b306336.txt">Table of n, a(n) for n = 0..440</a>

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

%F a(n) ~ -2*(-1)^n * n! * exp(3*Pi*n/2) / (exp(3*Pi/2) - 1)^(n+1). - _Vaclav Kotesovec_, Feb 09 2019

%p 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

%t nmax = 24; CoefficientList[Series[Sec[Log[1 + x]] + Tan[Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!

%t 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}]

%o (Python)

%o from itertools import accumulate

%o from sympy.functions.combinatorial.numbers import stirling

%o def A306336(n): # generator of terms

%o     if n == 0: return 1

%o     blist, c = (0,1), 0

%o     for k in range(1,n+1):

%o         c += stirling(n,k,kind=1,signed=True)*blist[-1]

%o         blist = tuple(accumulate(reversed(blist),initial=0))

%o     return c # _Chai Wah Wu_, Apr 18 2023

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

%K sign

%O 0,5

%A _Ilya Gutkovskiy_, Feb 08 2019