A390527 - OEIS

%I #15 Nov 12 2025 08:31:07

%S 1,2,8,35,157,712,3248,14869,68222,313487,1441999,6637879,30572032,

%T 140860379,649202036,2992721902,13798302512,63626933527,293426802548,

%U 1353295096618,6241829542241,28790709824587,132803515725281,612604408102432,2825929675569155

%N a(n) = Sum_{k=0..n} (4*k+1) * binomial(2*n+2*k+1,n-k)/(2*n+2*k+1).

%H Vincenzo Librandi, <a href="/A390527/b390527.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: g/(1-x*g^4) where g = 1+x*g^2 is the g.f. of A000108.

%t Table[Sum[(4*k+1)*Binomial[2*n+2*k+1,n-k]/(2*n+2*k+1),{k,0,n}],{n,0,22}] (* _Vincenzo Librandi_, Nov 12 2025 *)

%o (PARI) a(n) = sum(k=0, n, (4*k+1)*binomial(2*n+2*k+1, n-k)/(2*n+2*k+1));

%o (Magma) [&+[(4*k+1)*Binomial(2*n+2*k+1, n-k)/(2*n+2*k+1): k in [0..n]] : n in [0..30] ]; // _Vincenzo Librandi_, Nov 12 2025

%Y Cf. A026726, A390528.

%Y Cf. A000108.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Nov 09 2025