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A375381 E.g.f.: (exp(-x) - exp(x) - 2)/(exp(-x) + exp(x) - 4).

%I A375381 #11 Sep 02 2024 12:57:40
%S A375381 1,1,1,4,7,46,121,1114,3907,46246,202741,2933074,15430207,263817646,
%T A375381 1619195761,31943268634,224061282907,5009616448246,39531606447181,
%U A375381 987840438629794,8661323866026007,239217148602642046,2307185279184885001,69790939492563608554
%N A375381 E.g.f.: (exp(-x) - exp(x) - 2)/(exp(-x) + exp(x) - 4).
%F A375381 a(n) = n! * [x^n] (1 + sinh(x))/(2 - cosh(x)).
%F A375381 a(n) = Sum_{j=0..n-1,2} binomial(n, j) * a(j) for n > 0, a(0) = 1. (Note that the sum runs in steps of 2.)
%F A375381 a(n) ~ n! * (1 + 1/sqrt(3) + (-1)^n * (-1 + 1/sqrt(3))) / log(2 + sqrt(3))^(n+1). - _Vaclav Kotesovec_, Sep 02 2024
%p A375381 a := proc(n) option remember; local j;
%p A375381 ifelse(n = 0, 1, add(binomial(n, j) * a(j), j = 0..n-1, 2)) end:
%p A375381 # Or:
%p A375381 gf := (exp(-x) - exp(x) - 2)/(exp(-x) + exp(x) - 4):
%p A375381 series(gf, x, 24): seq(n!*coeff(%, x, n), n = 0..23);
%Y A375381 Cf. A094088 (even bisection), A331978 (odd bisection).
%K A375381 nonn
%O A375381 0,4
%A A375381 _Peter Luschny_, Aug 25 2024