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

%I A306336 #14 Apr 19 2023 09:04:30
%S A306336 1,1,0,1,-2,10,-50,320,-2340,19640,-184900,1932500,-22187200,
%T A306336 277576000,-3757884000,54732418000,-853278998000,14176686784000,
%U A306336 -250046057846000,4665989766386000,-91838330641200000,1901405069222360000,-41307212202493120000,939523370329035440000,-22327292561388519640000
%N A306336 Expansion of e.g.f. sec(log(1 + x)) + tan(log(1 + x)).
%H A306336 Vaclav Kotesovec, <a href="/A306336/b306336.txt">Table of n, a(n) for n = 0..440</a>
%F A306336 a(n) = Sum_{k=0..n} Stirling1(n,k)*A000111(k).
%F A306336 a(n) ~ -2*(-1)^n * n! * exp(3*Pi*n/2) / (exp(3*Pi/2) - 1)^(n+1). - _Vaclav Kotesovec_, Feb 09 2019
%p A306336 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 A306336 nmax = 24; CoefficientList[Series[Sec[Log[1 + x]] + Tan[Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
%t A306336 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 A306336 (Python)
%o A306336 from itertools import accumulate
%o A306336 from sympy.functions.combinatorial.numbers import stirling
%o A306336 def A306336(n): # generator of terms
%o A306336     if n == 0: return 1
%o A306336     blist, c = (0,1), 0
%o A306336     for k in range(1,n+1):
%o A306336         c += stirling(n,k,kind=1,signed=True)*blist[-1]
%o A306336         blist = tuple(accumulate(reversed(blist),initial=0))
%o A306336     return c # _Chai Wah Wu_, Apr 18 2023
%Y A306336 Cf. A000111, A003708, A009007, A048994, A317022.
%K A306336 sign
%O A306336 0,5
%A A306336 _Ilya Gutkovskiy_, Feb 08 2019