A026001 a(n) = T(3n,n), where T = Delannoy triangle (A008288).

1, 7, 85, 1159, 16641, 246047, 3707509, 56610575, 872893441, 13560999991, 211939849045, 3328419072535, 52481589415425, 830317511708367, 13174519143904245, 209559710593266719, 3340604559333629953, 53354776911196959335, 853607938952248383829, 13677336690921351929767

Offset

0,2

Comments

If the Delannoy triangle is defined by the Maple code in A008288, this is A008288(n, 3*n-2), n >= 1. - N. J. A. Sloane, Oct 29 2006

Links

Seiichi Manyama, Table of n, a(n) for n = 0..823

Lin Yang, Yu-Yuan Zhang, and Sheng-Liang Yang, The halves of Delannoy matrix and Chung-Feller properties of the m-Schröder paths, Linear Alg. Appl. (2024).

Formula

G.f.: F(G^(-1)(x)) where F = (2-t)/(3*t^2-8*t+2) and G = t*(t-1)^3/(t-2). - Mark van Hoeij, Oct 30 2011

From Peter Bala, Jan 29 2020: (Start)

a(n) = Sum_{k = 0..n} C(n,k)*C(3*n+k,n).

a(n) = Sum_{k = 0..n} C(n,k)*C(4*n-k,n).

a(n) = Sum_{k = 0..n} C(3*n,n-k)*C(3*n+k,k).

a(n) = Sum_{k = 0..n} 2^k*C(n,k)*C(3*n,k).

a(n) = Sum_{k = 0..n} C(4*n-k,k)*C(4*n-2*k,n-k).

3*n*(3*n - 1)*(3*n - 2)*(70*n^2 - 189*n + 127)*a(n) = 2*(15610*n^5 - 65562*n^4 + 102255*n^3 - 72864*n^2 + 23369*n - 2640)*a(n-1) - 3*(n - 1)* (3*n - 4)*(3*n - 5)*(70*n^2 - 49*n + 8)*a(n-2) with a(0) = 1, a(1) = 7.

(End)

a(n) = binomial(4*n, n)*hypergeom([-3*n, -n], [-4*n], -1). - Peter Luschny, Jan 31 2020

a(n) ~ sqrt(1 + 13/(4*sqrt(10))) * (223 + 70*sqrt(10))^n / (sqrt(Pi*n) * 3^(3*n + 1/2)). - Vaclav Kotesovec, Feb 13 2021

D-finite with recurrence +435*n*(3*n-1)*(3*n-2)*a(n) +(-53978*n^3+43545*n^2+39923*n-35580)*a(n-1) +3*(-57648*n^3+321915*n^2-580787*n+339980)*a(n-2) +9*(1634*n^3-11365*n^2+27137*n-22546)*a(n-3) -27*(3*n-10)*(3*n-11)*(n-3)*a(n-4)=0. - R. J. Mathar, Aug 01 2022

Maple

F := (2-t)/(3*t^2-8*t+2);  G := t*(t-1)^3/(t-2);  Ginv := RootOf(numer(G-x), t); ogf := series(eval(F, t=Ginv), x=0, 25); # Mark van Hoeij, Oct 30 2011

Mathematica

a[n_] := Binomial[4 n, n] Hypergeometric2F1[-3 n, -n, -4 n, -1];
Array[a, 20, 0] (* Peter Luschny, Jan 31 2020 *)

Crossrefs

Cf. A008288, A001850, A026000, A331329.

Sequence in context: A126344 A193578 A309187 * A371363 A064089 A216698

Adjacent sequences: A025998 A025999 A026000 * A026002 A026003 A026004

Keyword

nonn,easy

Author

Clark Kimberling

Extensions

Corrected and extended by N. J. A. Sloane, Oct 29 2006

Status

approved