A379025 - OEIS

%I #27 Dec 30 2024 02:20:45

%S 1,6,78,1149,17850,285711,4661727,77086008,1287322866,21661521945,

%T 366687839133,6237631866417,106535632157643,1825763898882189,

%U 31379978657609100,540688387589377764,9336602657251874754,161534120354250452361,2799488717098336992687

%N a(n) = Sum_{k=0..n} binomial(3*n+k-1,k) * binomial(3*n+k,n-k).

%H Vincenzo Librandi, <a href="/A379025/b379025.txt">Table of n, a(n) for n = 0..500</a>

%F a(n) = [x^n] ( (1 + x)/(1 - x - x^2) )^(3*n).

%F a(n) = binomial(3*n, n)*hypergeom([-n, 3*n, 1 + 3*n], [1/2 + n, 1 + n], -1/4). - _Stefano Spezia_, Dec 22 2024

%t Table[Sum[Binomial[3*n+k-1,k]*Binomial[3*n+k, n-k],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Dec 21 2024 *)

%t a[n_]:= Binomial[3*n, n]*HypergeometricPFQ[{-n, 3*n, 1 + 3*n}, {1/2 + n, 1 + n}, -1/4]; Array[a,19,0] (* _Stefano Spezia_, Dec 22 2024 *)

%o (PARI) a(n) = sum(k=0, n, binomial(3*n+k-1, k)*binomial(3*n+k, n-k));

%o (Magma) [&+[Binomial(3*n+k-1, k)*Binomial(3*n+k, n-k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Dec 21 2024

%Y Cf. A262910, A379022, A379026.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 14 2024