A339240 a(n) = n*2^(2*n-2) + n*binomial(2*n,n)/2.

0, 2, 14, 78, 396, 1910, 8916, 40684, 182552, 808614, 3545220, 15414212, 66556584, 285707708, 1220340296, 5189913240, 21988512304, 92850096902, 390913863012, 1641450064084, 6876023427080, 28741451864916, 119902111845208, 499304732388968, 2075821104461136, 8617006998238300

Offset: 0

Author

Michel Marcus, Nov 28 2020

Keywords

• nonn

Links

• Horst Alzer and Helmut Prodinger, Identities and Inequalities for Sums Involving Binomial Coefficients, INTEGERS 20 (2020) A9.

Crossrefs

Cf. A002697, A002457.

Programs

Mathematica

a[n_] := n*(2^(2*n - 2) + Binomial[2*n, n]/2); Array[a, 26, 0] (* Amiram Eldar, Nov 28 2020 *)

PARI

a(n) = n*2^(2*n-2) + n*binomial(2*n,n)/2;

PARI

a(n) = sum(k=0, n, binomial(n,k)*k*sum(j=0, k, binomial(n, j)));

Formula

a(n) = Sum_{k=0..n} binomial(n, k)*k*Sum_{j=0..k} binomial(n, j).

a(n) = A002697(n) + A002457(n-1), for n>0.

G.f.: x*(1/(1 - 4*x)^2 + 1/(1 - 4*x)^(3/2)). - Stefano Spezia, Nov 28 2020