A390721 - OEIS

A390721

a(n) = Sum_{k=0..n} (k+2) * binomial(5*n-4*k+2,n-k)/(5*n-4*k+2).

%I #13 Nov 17 2025 00:05:27

%S 1,3,15,103,833,7381,69276,676555,6802814,69949499,732099536,

%T 7773188813,83521931171,906478836870,9922793839711,109426668183175,

%U 1214551788118680,13557416935589156,152099318341418174,1714087810020632151,19395280422854560090,220265616659410691342

%N a(n) = Sum_{k=0..n} (k+2) * binomial(5*n-4*k+2,n-k)/(5*n-4*k+2).

%H Vincenzo Librandi, <a href="/A390721/b390721.txt">Table of n, a(n) for n = 0..920</a>

%F G.f.: g^2/(1-x*g) where g = 1+x*g^5 is the g.f. of A002294.

%t Table[Sum[(k+2)*Binomial[5*n-4*k+2,n-k]/(5*n-4*k+2),{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Nov 16 2025 *)

%o (PARI) a(n) = sum(k=0, n, (k+2)*binomial(5*n-4*k+2, n-k)/(5*n-4*k+2));

%o (Magma) [&+[(k+2)*Binomial(5*n-4*k+2, n-k)/(5*n-4*k+2): k in [0..n]] : n in [0..30] ]; // _Vincenzo Librandi_, Nov 16 2025

%Y Cf. A000245, A390719, A390720.

%Y Cf. A002294.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Nov 16 2025