%I #13 Sep 22 2025 16:01:31

%S 1,2,5,16,57,211,793,3004,11441,43759,167961,646647,2496145,9657701,

%T 37442161,145422676,565722721,2203961431,8597496601,33578000611,

%U 131282408401,513791607421,2012616400081,7890371113951,30957699535777,121548660036301,477551179875953

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

%H G. C. Greubel, <a href="/A323229/b323229.txt">Table of n, a(n) for n = 0..1000</a>

%F 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).

%F a(n) = A212382(2*n, n). - _Alois P. Heinz_, May 03 2019

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

%p gf := (1-3*x)/(2*(x-1)*x) + (I*(1-2*x))/(2*x*sqrt(4*x-1));

%p ser := series(gf, x, len+2):

%p seq(coeff(ser, x, n), n=0..len) end: aList(27);

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

%o (Magma) [Binomial(2*n, n+1) + 1: n in [0..30]]; // _G. C. Greubel_, Dec 26 2021

%o (SageMath) [binomial(2*n, n+1) + 1 for n in (0..30)] # _G. C. Greubel_, Dec 26 2021

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

%Y Cf. A212382.

%K nonn,easy

%O 0,2

%A _Peter Luschny_, Feb 12 2019