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%I #36 Jan 09 2024 11:08:56
%S 1,7,85,1159,16641,246047,3707509,56610575,872893441,13560999991,
%T 211939849045,3328419072535,52481589415425,830317511708367,
%U 13174519143904245,209559710593266719,3340604559333629953,53354776911196959335,853607938952248383829,13677336690921351929767
%N a(n) = T(3n,n), where T = Delannoy triangle (A008288).
%C 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
%H Seiichi Manyama, <a href="/A026001/b026001.txt">Table of n, a(n) for n = 0..823</a>
%H Lin Yang, Yu-Yuan Zhang, and Sheng-Liang Yang, <a href="https://doi.org/10.1016/j.laa.2023.12.021">The halves of Delannoy matrix and Chung-Feller properties of the m-Schröder paths</a>, Linear Alg. Appl. (2024).
%F 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
%F From _Peter Bala_, Jan 29 2020: (Start)
%F a(n) = Sum_{k = 0..n} C(n,k)*C(3*n+k,n).
%F a(n) = Sum_{k = 0..n} C(n,k)*C(4*n-k,n).
%F a(n) = Sum_{k = 0..n} C(3*n,n-k)*C(3*n+k,k).
%F a(n) = Sum_{k = 0..n} 2^k*C(n,k)*C(3*n,k).
%F a(n) = Sum_{k = 0..n} C(4*n-k,k)*C(4*n-2*k,n-k).
%F 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.
%F (End)
%F a(n) = binomial(4*n, n)*hypergeom([-3*n, -n], [-4*n], -1). - _Peter Luschny_, Jan 31 2020
%F 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
%F 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
%p 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
%t a[n_] := Binomial[4 n, n] Hypergeometric2F1[-3 n, -n, -4 n, -1];
%t Array[a, 20, 0] (* _Peter Luschny_, Jan 31 2020 *)
%Y Cf. A008288, A001850, A026000, A331329.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_
%E Corrected and extended by _N. J. A. Sloane_, Oct 29 2006
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