A306609 a(n) = Sum_{k=0..n} k*binomial(4*n+2,2*k).

0, 15, 465, 11102, 236997, 4751010, 91474890, 1712391420, 31398038701, 566621243642, 10097483539038, 178113001428004, 3115342162844450, 54103694774702292, 933929099838928692, 16037182307150776056, 274132978890654857853, 4667160114821964359530, 79177297937966956038102, 1338972240005810710258452

Offset: 0

Author

Robert Israel, Feb 28 2019

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..828
• MathOverflow, Specific partial sum of even/odd binomial coefficients

Crossrefs

Cf. A000346.

Programs

GAP

List([0..30],n->Sum([0..n],k->k*Binomial(4*n+2,2*k))); # Muniru A Asiru, Mar 01 2019

Maple

f:= n -> (n+1/2)*(16^n-binomial(4*n,2*n)):
map(f, [$0..30]);

Mathematica

Table[Sum[k Binomial[4n+2,2k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jun 14 2024 *)

Formula

a(n) = (n+1/2)*(16^n - binomial(4*n,2*n)) = (2*n+1)*A000346(2*n-1).

-512*(4*n + 1)*(86*n + 213)*(3 + 4*n)*(n + 1)*a(n) + 32*(2336*n^4 + 8800*n^3 + 10524*n^2 + 11540*n + 9703)*a(n + 1) - 2*(n + 2)*(544*n^3 - 1072*n^2 + 1138*n + 8055)*a(n + 2) - (2*n + 5)*(26*n - 31)*(n + 3)*(n + 2)*a(n + 3) = 0.

a(n) ~ 16^n * (n - sqrt(n/(2*Pi)) + 1/2).