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

0,3

Comments

Partial sums are A129384.

Links

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

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)).

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 *)

Prog

(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
(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

Crossrefs

Cf. A001405, A129384.

Sequence in context: A355640 A370586 A276425 * A052957 A275441 A197465

Adjacent sequences: A129380 A129381 A129382 * A129384 A129385 A129386

Keyword

easy,nonn

Author

Paul Barry, Apr 12 2007

Status

approved