A120260 Diagonal sums of number triangle A120258.

1, 1, 2, 3, 8, 24, 92, 432, 2740, 23822, 264185, 3545166, 59474514, 1343942004, 41179884383, 1593533376361, 74665098131246, 4404743069577837, 351138858279113987, 37740395752334771775, 5093113605218543006445

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Formula

a(n)=sum{k=0..floor(n/2), Product{j=0..k-1, C(2n-4k+j, n-2k)/C(n-2k+j, j)}}

Limit_{n->oo} a(n)^(1/n^2) = r^(r^2/2) * (2-3*r)^((2-3*r)^2/2) / (2^(2*(1-2*r)^2) * (1-r)^((1-r)^2) * (1-2*r)^((1-2*r)^2)) = 1.133380884076924860904704854418..., where r = 0.201760656726887011996310570327419178... is the root of the equation 2^(8-16*r) * (2-3*r)^(-6+9*r) * (1-2*r)^(4-8*r) * (1-r)^(2-2*r) * r^r = 1. - Vaclav Kotesovec, Aug 29 2023

Mathematica

Table[Sum[Product[Binomial[2*n-4*k+j, n-2*k]/Binomial[n-2*k+j, j], {j, 0, k-1}], {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 29 2023 *)
Table[Sum[BarnesG[1 + k] * BarnesG[2 - 2*k + n]^2 * BarnesG[1 - 3*k + 2*n] * Gamma[1 - 4*k + 2*n] / (BarnesG[1 - k + n]^2 * BarnesG[2 - 4*k + 2*n] * Gamma[1 - 2*k + n]^2), {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 29 2023 *)

Crossrefs

Sequence in context: A038561 A055981 A182212 * A202592 A002619 A286820

Adjacent sequences: A120257 A120258 A120259 * A120261 A120262 A120263

Keyword

easy,nonn

Author

Paul Barry, Jun 13 2006

Status

approved