A323229 a(n) = binomial(2*n, n+1) + 1.

1, 2, 5, 16, 57, 211, 793, 3004, 11441, 43759, 167961, 646647, 2496145, 9657701, 37442161, 145422676, 565722721, 2203961431, 8597496601, 33578000611, 131282408401, 513791607421, 2012616400081, 7890371113951, 30957699535777, 121548660036301, 477551179875953

Offset: 0

Author

Peter Luschny, Feb 12 2019

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A323230 (d=0), A260878 (d=1), this sequence (d=2).

Cf. A212382.

Programs

Magma

[Binomial(2*n, n+1) + 1: n in [0..30]]; // G. C. Greubel, Dec 26 2021

Maple

aList := proc(len) local gf, ser; assume(Im(x) > 0);
gf := (1-3*x)/(2*(x-1)*x) + (I*(1-2*x))/(2*x*sqrt(4*x-1));
ser := series(gf, x, len+2):
seq(coeff(ser, x, n), n=0..len) end: aList(27);

Mathematica

Table[Binomial[2n, n+1] + 1, {n, 0, 26}]

Sage

[binomial(2*n, n+1) + 1 for n in (0..30)] # G. C. Greubel, Dec 26 2021

Formula

Let G(x) = (1-3*x)/(2*(x-1)*x) + (I*(1-2*x))/(2*x*sqrt(4*x-1)) with Im(x) > 0, then a(n) = [x^n] G(x). The generating function G(x) satisfies the differential equation 6*x^3 - 4*x + 1 = (8*x^5 - 22*x^4 + 21*x^3 - 8*x^2 + x)*diff(G(x), x) + (4*x^4 - 14*x^3 + 17*x^2 - 8*x + 1)*G(x).

a(n) = A212382(2*n, n). - Alois P. Heinz, May 03 2019