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

1, 1, 1, 4, 7, 46, 121, 1114, 3907, 46246, 202741, 2933074, 15430207, 263817646, 1619195761, 31943268634, 224061282907, 5009616448246, 39531606447181, 987840438629794, 8661323866026007, 239217148602642046, 2307185279184885001, 69790939492563608554

Offset: 0

Author

Peter Luschny, Aug 25 2024

Keywords

• nonn

Crossrefs

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

Programs

Maple

a := proc(n) option remember; local j;
ifelse(n = 0, 1, add(binomial(n, j) * a(j), j = 0..n-1, 2)) end:
# Or:
gf := (exp(-x) - exp(x) - 2)/(exp(-x) + exp(x) - 4):
series(gf, x, 24): seq(n!*coeff(%, x, n), n = 0..23);

Formula

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

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.)

a(n) ~ n! * (1 + 1/sqrt(3) + (-1)^n * (-1 + 1/sqrt(3))) / log(2 + sqrt(3))^(n+1). - Vaclav Kotesovec, Sep 02 2024