A129369 Expansion of 1/sqrt(1-4*x) - x/sqrt(1-4*x^2).

1, 1, 6, 18, 70, 246, 924, 3412, 12870, 48550, 184756, 705180, 2704156, 10399676, 40116600, 155114088, 601080390, 2333593350, 9075135300, 35345215180, 137846528820, 538257689684, 2104098963720, 8233430022168, 32247603683100

Offset

0,3

Comments

Partial sums are A129368.

Links

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

Formula

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

From G. C. Greubel, Jan 31 2024: (Start)

a(2*n) = A001448(n).

a(2*n+1) = (n+1)*( 2*A000108(2*n+1) - A000108(n) ).

a(2*n+1) = 2*A002458(n) - A000984(n).

(n+1)^3*(n+2)*(n+3)*a(n+3) = 2*(n+1)*(n+2)^3*(2*n+1)*a(n+2) + 4*(n+1)^4*(n+3)*a(n+1) - 8*n*(n+2)^3*(2*n+1)*a(n), with a(0)=a(1) = 1, a(2) = 6. (End)

Mathematica

CoefficientList[Series[1/Sqrt[1-4x]-x/Sqrt[1-4x^2], {x, 0, 30}], x] (* Harvey P. Dale, Feb 02 2012 *)

Prog

(Magma) B:=Binomial; [B(2*n, n) - (n mod 2)*B(n-1, Floor((n-1)/2)): n in [0..60]]; // G. C. Greubel, Jan 31 2024
(SageMath) [binomial(2*n, n) - (n%2)*binomial(n-1, (n-1)//2) for n in range(61)] # G. C. Greubel, Jan 31 2024

Crossrefs

Cf. A000108, A000984, A001448, A002458, A129368.

Sequence in context: A200151 A000623 A304941 * A095853 A027266 A242278

Adjacent sequences: A129366 A129367 A129368 * A129370 A129371 A129372

Keyword

easy,nonn

Author

Paul Barry, Apr 11 2007

Status

approved