A378405 - OEIS

%I #9 Nov 25 2024 07:35:39

%S 1,1,3,13,47,171,651,2507,9703,37831,148393,584673,2312267,9174179,

%T 36500257,145566333,581746503,2329206823,9341025429,37516150599,

%U 150874376997,607479424817,2448608334087,9879562243961,39897969991075,161260133795371

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

%F a(n) = [x^n] (1 + x + x^2 * (1 + x)^2)^n.

%F a(n) ~ sqrt(c) * d^n / sqrt(Pi*n), where d = 4.1236218756427610008124277125077732535524468472302771364162049292... is the greatest root of the equation 31 - 256*d + 30*d^2 - 104*d^3 + 27*d^4 = 0 and c = 0.3580097056143148810957018324419771260252367819271313726816... is the positive real root of the equation -31 - 3024*c + 51376*c^2 - 257536*c^3 + 386304*c^4 = 0. - _Vaclav Kotesovec_, Nov 25 2024

%t Table[Sum[Binomial[n, k]*Binomial[n + k, n - 2*k], {k, 0, n/2}], {n, 0, 30}] (* _Vaclav Kotesovec_, Nov 25 2024 *)

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

%Y Cf. A005725, A378406.

%Y Cf. A001850, A082759.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Nov 25 2024