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%I #16 May 27 2022 08:10:26
%S 0,4,137,2136,23452,209840,1640346,11648224,76976048,481048128,
%T 2874897670,16564931504,92584313112,504313834336,2687067833492,
%U 14045889333120,72202366588096,365713117287680,1828223537042142,9032706189007888,44158716127799240,213826835772518304
%N a(n) = S2(n,3), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.
%H G. C. Greubel, <a href="/A089666/b089666.txt">Table of n, a(n) for n = 0..1000</a>
%H Jun Wang and Zhizheng Zhang, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00206-1">On extensions of Calkin's binomial identities</a>, Discrete Math., 274 (2004), 331-342.
%F a(n) = n*(15*n^3 + 30*n^2 + 21*n - 2)*4^(n-3) - (n-1)^2*n^2*(n+1)*binomial(2*n, n)/(8*(2*n-1)). (See Wang and Zhang, p. 338.)
%F From _G. C. Greubel_, May 25 2022: (Start)
%F a(n) = n*(15*n^3 + 30*n^2 + 21*n - 2)*4^(n-3) - 9*binomial(n+1, 3)^2 * Catalan(n- 1)/(n+1).
%F G.f.: x*(4*(1 + 15*x + 12*x^2 + 8*x^3) - 3*x*(1 + 6*x - 6*x^2 + 4*x^3)*sqrt(1-4*x))/(1-4*x)^5. (End)
%p S2:= (n, t) -> add(k^t*add(binomial(n, j), j = 0..k)^2, k = 0..n);
%p seq(S2(n, 3), n = 0..40);
%t Table[n*(15*n^3+30*n^2+21*n-2)*4^(n-3) -(n-1)^2*n^2*(n+1)*Binomial[2*n,n]/(8*(2*n -1)), {n, 0, 40}] (* _G. C. Greubel_, May 25 2022 *)
%o (SageMath) [n*(15*n^3+30*n^2+21*n-2)*4^(n-3) - 9*binomial(n+1, 3)^2 * catalan_number(n-1)/(n+1) for n in (0..40)] # _G. C. Greubel_, May 25 2022
%Y Sequences of S2(n, t): A003583 (t=0), A089664 (t=1), A089665 (t=2), this sequence (t=3), A089667 (t=4), A089668 (t=5).
%Y Cf. A000108, A089658, A089669.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Jan 04 2004
%E Name changed by _G. C. Greubel_, May 25 2022
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