A324568 a(n) = Sum_{i=0..n, j=0..n} (binomial(2*i, j) + binomial(2*j, i)).

2, 8, 32, 124, 482, 1882, 7380, 29036, 114530, 452638, 1791638, 7100430, 28167986, 111837902, 444351292, 1766536044, 7026526226, 27960911422, 111308958942, 443258277254, 1765690504666, 7035402933402, 28039342445582, 111773962249054, 445654589001882

Offset: 0

Author

Vaclav Kotesovec, Mar 07 2019

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..1653

Crossrefs

Cf. A000984, A006134, A007685, A324566, A324567.

Programs

Maple

R:= 2: r:= 2;
for n from 1 to 30 do
  v:= 2*binomial(2*n,n) + 2*add(binomial(2*n,j),j=0..n-1) + 2*add(binomial(2*j,n),j=ceil(n/2) .. n-1);
  r:= r+v;
  R:= R,r;
od:
R; # Robert Israel, Mar 02 2025

Mathematica

Table[Sum[Binomial[2*i, j] + Binomial[2*j, i], {i, 0, n}, {j, 0, n}], {n, 0, 30}]

Formula

Recurrence: 2*(n+1)*(5*n^2 - 21*n + 20)*a(n) = (85*n^3 - 332*n^2 + 243*n + 100)*a(n-1) - 3*(65*n^3 - 298*n^2 + 377*n - 100)*a(n-2) + 2*(20*n^3 - 109*n^2 + 191*n - 100)*a(n-3) + 8*(2*n - 5)*(5*n^2 - 11*n + 4)*a(n-4).

a(n) ~ 4^(n+1)/3 * (1 + 5/(3*sqrt(Pi*n))).