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
From Seiichi Manyama, Sep 13 2025: (Start)
a(n) = [x^n] (1-x)^n/(1-2*x)^(3*n+1).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(n,k) * binomial(3*n+k,k). (End)
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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Corrected and extended by N. J. A. Sloane, Oct 29 2006
STATUS
approved