A358446 a(n) = n! * Sum_{k=0..floor(n/2)} 1/binomial(n-k, k).

1, 1, 4, 9, 56, 190, 1704, 7644, 93120, 516240, 8136000, 53523360, 1047548160, 7961241600, 187132377600, 1611967392000, 44311886438400, 426483893606400, 13428757601280000, 142790947407360000, 5066854992138240000, 58981696577556480000, 2328441680297779200000

Offset: 0

Author

Vladimir Kruchinin, Nov 16 2022

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..449

Crossrefs

Cf. A003149, A143216, A344391.

Programs

Maple

egf := (2*x+1)/((x-1)*(x+1)*(x^2-x-1))-(x*log((1-x)^2*(x+1)))/(-x^2+x+1)^2:
ser := series(egf, x, 22): seq(n!*coeff(ser, x, n), n = 0..20); # Peter Luschny, Nov 17 2022

Maxima

a(n):=factorial(n)*sum(1/binomial(n-k,k),k,0,floor(n/2));

SageMath

def A358446(n):
    return sum(A143216(n, k) // A344391(n, k) for k in range((n+2)//2))
print([A358446(n) for n in range(23)]) # Peter Luschny, Nov 17 2022

Formula

E.g.f.: (2*x+1)/((x-1)*(x+1)*(x^2-x-1))-(x*log((1-x)^2*(x+1)))/(-x^2+x+1)^2.

a(n) ~ n! * (3 + (-1)^n)/2. - Vaclav Kotesovec, Nov 17 2022

a(n) = Sum_{k=0..floor(n/2)} A143216(n, k)/A344391(n, k). - Peter Luschny, Nov 17 2022