A342967 a(n) = 1 + Sum_{j=1..n} Product_{k=0..j-1} binomial(2*n-1,n+k) / binomial(2*n-1,k).

1, 2, 5, 22, 177, 2606, 70226, 3457742, 311348897, 51177188350, 15377065068510, 8430169458379450, 8446194335222422950, 15435904380166258833482, 51546769958534244310727102, 313937270864810066000897492222, 3493348088919874482660174997662017

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..87

Formula

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

a(n) ~ c * exp(1/12) * 2^(4*n^2 - 1/12) / (A * n^(1/12) * 3^(9*n^2/4 - 1/6)), where c = JacobiTheta3(0,1/3) = EllipticTheta[3, 0, 1/3] = 1.69145968168171534134842... if n is even, and c = JacobiTheta2(0,1/3) = EllipticTheta[2, 0, 1/3] = 1.69061120307521423305296... if n is odd, and A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 02 2021

Mathematica

a[n_] := 1 + Sum[Product[Binomial[2*n - 1, n + k]/Binomial[2*n - 1, k], {k, 0, j - 1}], {j, 1, n}]; Array[a, 17, 0] (* Amiram Eldar, Apr 01 2021 *)
Table[1 + BarnesG[2*n + 1] * Sum[BarnesG[j + 1]*BarnesG[n - j + 1] / (BarnesG[n + j + 1]*BarnesG[2*n - j + 1]), {j, 1, n}], {n, 0, 15}] (* Vaclav Kotesovec, Apr 02 2021 *)

Prog

(PARI) a(n) = 1+sum(j=1, n, prod(k=0, j-1, binomial(2*n-1, n+k)/binomial(2*n-1, k)));
(PARI) a(n) = sum(j=0, n, prod(k=0, n-1, binomial(n+k, j)/binomial(j+k, j)));

Crossrefs

Row sums of A342972.

Cf. A001181, A005362, A005363, A005364, A005365, A005366, A116925.

Sequence in context: A067549 A262842 A361331 * A042933 A005203 A193660

Adjacent sequences: A342964 A342965 A342966 * A342968 A342969 A342970

Keyword

nonn

Author

Seiichi Manyama, Apr 01 2021

Status

approved