A396199 - OEIS

%I #14 May 19 2026 16:39:58

%S 1,6,18,52,130,324,756,1768,3978,8980,19756,43608,94484,205352,440040,

%T 945616,2009434,4281012,9037692,19123960,40160316,84514936,176713048,

%U 370203312,771256900,1609622664,3343062456,6954560368,14405875048,29885491920,61763349968,127821667232

%N a(n) = (n + 1)*(2^(n + 1) - binomial(n, floor(n / 2))), row sums of A396200.

%H Paolo Xausa, <a href="/A396199/b396199.txt">Table of n, a(n) for n = 0..1000</a>

%F From _Vaclav Kotesovec_, May 19 2026: (Start)

%F Recurrence: (n-1)*n*a(n) = 2*(n-1)*(n+1)*a(n-1) + 4*(n-2)*n*a(n-2) - 8*(n-1)*n*a(n-3).

%F a(n) ~ n * 2^(n+1).

%F a(n) = (n+1)*2^(n+1) - 4*(n+1)*Gamma(n)/(n*Gamma(n/2)^2) if n is even and n>0.

%F a(n) = (n+1)*2^(n+1) - 2*n*Gamma(n)/Gamma((n+1)/2)^2 if n is odd. (End)

%p a := n -> (n + 1)*(2^(n + 1) - binomial(n, iquo(n, 2))):

%p seq(a(n), n = 0..31);

%t Table[Sum[Sum[Binomial[2*i, i] * Binomial[n-2*i, k-i], {i, 0, n}], {k, 0, n}], {n, 0, 30}] (* _Vaclav Kotesovec_, May 19 2026 *)

%t (* Alternative: *)

%t A396199[n_] := (n + 1)*(2^(n + 1) - Binomial[n, Quotient[n, 2]]);

%t Array[A396199, 35, 0] (* _Paolo Xausa_, May 19 2026 *)

%Y Cf. A396200.

%K nonn

%O 0,2

%A _Peter Luschny_, May 19 2026