OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
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
MAPLE
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));
MATHEMATICA
Table[Binomial[2 n + 2, n + 2] - 1, {n, 0, 25}]
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Luschny, Feb 13 2019
STATUS
approved