0,2

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

Let G(x) = (2*x^2-x+1)/(2*(x-1)*x^2)-(I*(2*x-1))/(2*x^2*sqrt(4*x-1)) with Im(x) > 0, then a(n) = [x^n] G(x). The generating function G(x) satisfies the differential equation 9*x - 16*x^2 + 4*x^3 = (8*x^5 - 22*x^4 + 21*x^3 - 8*x^2 + x)*diff(G(x), x) + (12*x^4 - 36*x^3 + 38*x^2 - 16*x + 2)*G(x).

From Peter Bala, Feb 25 2022: (Start)

a(n) = Sum_{k = 0..n+1} binomial(n+k,k+1).

a(n) = Sum_{k = 0..n-1} binomial(n+k+2,k+1).

More generally, Sum_{k = 0..n+m} binomial(n+k,k+1) = Sum_{k = 0..n-1} binomial(n+k+m+1,k+1) = binomial(2*n+m+1,n) - 1. (End)

a(n) = A001791(n+1) - 1. - Hugo Pfoertner, Feb 26 2022

a(n) = n/(n+2) * binomial(2*n+2, n+1) * Sum_{k = 0..n+1} 1/binomial(n+k+1, k). - Peter Bala, Aug 05 2025

aList := proc(len) local gf, ser; assume(Im(x) > 0);

gf := (2*x^2 - x + 1)/(2*(x - 1)*x^2) - (I*(2*x - 1))/(2*x^2*sqrt(4*x - 1));

ser := series(gf, x, len+4):

seq(coeff(ser, x, n), n=0..len) end: lprint(aList(25));

Table[Binomial[2 n + 2, n + 2] - 1, {n, 0, 25}]

(Magma)

[Binomial(2*n+2, n+2) -1: n in [0..30]]; // G. C. Greubel, Apr 22 2024

(SageMath)

[binomial(2*n+2, n+2) - 1 for n in range(31)] # G. C. Greubel, Apr 22 2024

Cf. A001791, A014473, A030662 (d=0), A010763 (d=1), this sequence (d=2).

Sequence in context: A151540 A226466 A363964 * A026544 A026527 A037786

Adjacent sequences: A322935 A322936 A322937 * A322939 A322940 A322941

nonn,easy

Peter Luschny, Feb 13 2019

approved