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A342161 Expansion of the exponential generating function (tanh(x) - sech(x) + 1) * exp(x).

%I A342161 #23 Mar 06 2021 05:02:15
%S A342161 0,1,3,4,-3,-14,63,274,-1383,-7934,50523,353794,-2702763,-22368254,
%T A342161 199360983,1903757314,-19391512143,-209865342974,2404879675443,
%U A342161 29088885112834,-370371188237523,-4951498053124094,69348874393137903,1015423886506852354,-15514534163557086903
%N A342161 Expansion of the exponential generating function (tanh(x) - sech(x) + 1) * exp(x).
%H A342161 A. Randrianarivony and J. Zeng, <a href="http://dx.doi.org/10.1006/aama.1996.0001">Une famille de polynomes qui interpole plusieurs suites classiques de nombres</a>, Adv. Appl. Math. 17 (1996), 1-26. See Section 6 (zeroth column of matrix b_{n,k} on p. 19).
%F A342161 a(n) = A323834(n, 0).
%F A342161 a(n) = n! [x^n] (tanh(x) - sech(x) + 1) * exp(x).
%F A342161 a(n) = Sum_{i=1..n} binomial(n,i) * (-1)^floor((i-1)/2) * A000111(i).
%p A342161 series((2*exp(x)-2)/(exp(-2*x)+1),x,30):seq(n!*coeff(%,x,n),n=0..24); # _Peter Luschny_, Mar 05 2021
%o A342161 (PARI) my(x='x+O('x^30)); concat(0, Vec(serlaplace((-1/cosh(x) + tanh(x) + 1)*exp(x)))) \\ _Michel Marcus_, Mar 05 2021
%o A342161 (SageMath)
%o A342161 def A323834List(prec):
%o A342161     R.<x> = PowerSeriesRing(QQ, default_prec=prec)
%o A342161     f = (2*exp(2*x)*(exp(x) - 1))/(exp(2*x) + 1)
%o A342161     return f.egf_to_ogf().list()
%o A342161 print(A323834List(25)) # _Peter Luschny_, Mar 05 2021
%Y A342161 Cf. A000111, A323834.
%K A342161 sign
%O A342161 0,3
%A A342161 _Petros Hadjicostas_, Mar 03 2021