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a(n) = n*2^(2*n-2) + n*binomial(2*n,n)/2.
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%I #10 Nov 28 2020 20:08:53

%S 0,2,14,78,396,1910,8916,40684,182552,808614,3545220,15414212,

%T 66556584,285707708,1220340296,5189913240,21988512304,92850096902,

%U 390913863012,1641450064084,6876023427080,28741451864916,119902111845208,499304732388968,2075821104461136,8617006998238300

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

%H Horst Alzer and Helmut Prodinger, <a href="http://math.colgate.edu/~integers/u9/u9.mail.html">Identities and Inequalities for Sums Involving Binomial Coefficients</a>, INTEGERS 20 (2020) A9.

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

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

%F G.f.: x*(1/(1 - 4*x)^2 + 1/(1 - 4*x)^(3/2)). - _Stefano Spezia_, Nov 28 2020

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

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

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

%Y Cf. A002697, A002457.

%K nonn

%O 0,2

%A _Michel Marcus_, Nov 28 2020