A375381 - OEIS

%I #11 Sep 02 2024 12:57:40

%S 1,1,1,4,7,46,121,1114,3907,46246,202741,2933074,15430207,263817646,

%T 1619195761,31943268634,224061282907,5009616448246,39531606447181,

%U 987840438629794,8661323866026007,239217148602642046,2307185279184885001,69790939492563608554

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

%F a(n) = n! * [x^n] (1 + sinh(x))/(2 - cosh(x)).

%F 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 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 a := proc(n) option remember; local j;

%p ifelse(n = 0, 1, add(binomial(n, j) * a(j), j = 0..n-1, 2)) end:

%p # Or:

%p gf := (exp(-x) - exp(x) - 2)/(exp(-x) + exp(x) - 4):

%p series(gf, x, 24): seq(n!*coeff(%, x, n), n = 0..23);

%Y Cf. A094088 (even bisection), A331978 (odd bisection).

%K nonn

%O 0,4

%A _Peter Luschny_, Aug 25 2024