A391168 - OEIS

%I #16 Dec 04 2025 09:48:20

%S 1,3,10,39,172,827,4223,22513,123906,698800,4017611,23459981,

%T 138753411,829497070,5004289494,30428025015,186279229008,

%U 1147233165477,7102899516394,44184059607680,276013127801339,1730802569428854,10890903201239142,68745550922693495

%N Expansion of g/(1 - x*g)^2, where g = 1+x*g^3 is the g.f. of A001764.

%H Vincenzo Librandi, <a href="/A391168/b391168.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..n} (k+1)^2 * binomial(3*n-2*k+1,n-k)/(3*n-2*k+1).

%t Table[Sum[(k+1)^2*Binomial[3*n-2*k+1,n-k]/(3*n-2*k+1),{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Dec 03 2025 *)

%o (PARI) a(n) = sum(k=0, n, (k+1)^2*binomial(3*n-2*k+1, n-k)/(3*n-2*k+1));

%o (Magma) [&+[(k+1)^2 * Binomial(3*n-2*k+1,n-k)/(3*n-2*k+1): k in [0..n]] : n in [0..30] ]; // _Vincenzo Librandi_, Dec 03 2025

%Y Cf. A001764, A098746, A391169, A391170.

%Y Cf. A391173.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Dec 01 2025