A129383 Expansion of g(x) - x*g(x^2), where g(x) is the g.f. of A001405.

1, 0, 2, 2, 6, 8, 20, 32, 70, 120, 252, 452, 924, 1696, 3432, 6400, 12870, 24240, 48620, 92252, 184756, 352464, 705432, 1351616, 2704156, 5199376, 10400600, 20056584, 40116600, 77555328, 155117520, 300533760, 601080390, 1166790240

Offset: 0

Author

Paul Barry, Apr 12 2007

Keywords

• easy
• nonn

Comments

Partial sums are A129384.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A001405, A129384.

Programs

Magma

A129383:= func< n | Binomial(n,Floor(n/2)) - (n mod 2)*Binomial(Floor((n-1)/2),Floor((n-1)/4))  >;
[A129383(n): n in [0..40]]; // G. C. Greubel, Feb 03 2024

Mathematica

A129383[n_]:= With[{B=Binomial,F=Floor}, B[n,F[n/2]] - Mod[n,2]*B[(n- 1)/2, F[(n-1)/4]]];
Table[A129383[n], {n,0,40}] (* G. C. Greubel, Feb 03 2024 *)

SageMath

def A129383(n): return binomial(n,n//2) - (n%2)*binomial((n-1)/2,(n-1)//4)
[A129383(n) for n in range(41)] # G. C. Greubel, Feb 03 2024

Formula

G.f.: 2/(1-2*x+sqrt(1-4*x^2)) - 2*x/(1-2*x^2+sqrt(1-4*x^4)).

a(n) = binomial(n,floor(n/2)) - (1/2)*(1-(-1)^n)*binomial((n-1)/2, floor((n-1)/4)).