G. C. Greubel, <a href="/A129383/b129383_1.txt">Table of n, a(n) for n = 0..1000</a>

reviewed

approved

proposed

reviewed

editing

proposed

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

G. C. Greubel, <a href="/A129383/b129383_1.txt">Table of n, a(n) for n = 0..1000</a>

G.f.: 2/(1-2x2*x+sqrt(1-4x4*x^2)) -2x 2*x/(1-2x2*x^2+sqrt(1-4x^4)); a(n)=C(n,floor(n/2))-C((n-1)/2,floor((n-1)/*x^4))(1-(-1)^n)/2;.

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

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 *)

(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

Cf. A001405, A129384.

approved

editing

_Paul Barry (pbarry(AT)wit.ie), _, Apr 12 2007

Expansion of g(x)-xg(x^2), g(x) 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

0,3

Partial sums are A129384.

G.f.: 2/(1-2x+sqrt(1-4x^2))-2x/(1-2x^2+sqrt(1-4x^4)); a(n)=C(n,floor(n/2))-C((n-1)/2,floor((n-1)/4))(1-(-1)^n)/2;

easy,nonn

Paul Barry (pbarry(AT)wit.ie), Apr 12 2007

approved