%I #21 Jan 01 2026 14:16:48

%S 1,5,24,121,637,3468,19380,110561,641355,3771885,22439040,134796060,

%T 816540124,4982228488,30593078076,188908910769,1172307070375,

%U 7307391855855,45732182412840,287246023045905,1810153271613645,11441454994023600,72517477266539760

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

%H Vincenzo Librandi, <a href="/A390968/b390968.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: g^4 * (1 + x*g^2), where g = 1+x*g^3 is the g.f. of A001764.

%F a(n) = 2*A006629(n) - A001764(n+1) = (3*n+2) * (3*n+3)!/((n+2)! * (2*n+3)!).

%t Table[Sum[(k+1)^2*(2*k+1)*Binomial[3*n-k,n-k]/(2*n+1),{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Dec 05 2025 *)

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

%o (Magma) [&+[(k+1)^2*(2*k+1)*Binomial(3*n-k, n-k)/(2*n+1): k in [0..n]] : n in [0..30] ]; // _Vincenzo Librandi_, Dec 05 2025

%Y Cf. A001764, A006013, A390969, A390970.

%Y Cf. A006629, A070857, A000139, A013698.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Nov 25 2025