A390710 - OEIS

A390710

a(n) = Sum_{k=0..n} (5*k+1) * binomial(4*n+k+1,n-k)/(4*n+k+1).

%I #13 Dec 08 2025 04:28:58

%S 1,2,11,73,528,4017,31607,254779,2091647,17420270,146780121,

%T 1248679940,10708942769,92478844907,803403848506,7016088024023,

%U 61554363004544,542255447285265,4794512298020555,42532684423049866,378446187960196753,3376541608977259909

%N a(n) = Sum_{k=0..n} (5*k+1) * binomial(4*n+k+1,n-k)/(4*n+k+1).

%H Vincenzo Librandi, <a href="/A390710/b390710.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: g/(1 - x*g^5), where g = 1+x*g^4 is the g.f. of A002293.

%F G.f.: 1/(1 + 1/g - g), where g = 1+x*g^4 is the g.f. of A002293. - _Seiichi Manyama_, Dec 08 2025

%t Table[Sum[(5*k+1)*Binomial[4*n+k+1,n-k]/(4*n+k+1),{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Nov 16 2025 *)

%o (PARI) a(n) = sum(k=0, n, (5*k+1)*binomial(4*n+k+1, n-k)/(4*n+k+1));

%o (Magma) [&+[(5*k+1)*Binomial(4*n+k+1, n-k)/(4*n+k+1): k in [0..n]] : n in [0..30] ]; // _Vincenzo Librandi_, Nov 16 2025

%Y Cf. A069271, A107026, A390709.

%Y Cf. A002293.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Nov 15 2025