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A089668 a(n) = S2(n,5), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.

%I A089668 #10 May 26 2022 00:47:48
%S A089668 0,4,521,17136,320716,4356560,48024786,456843520,3893995184,
%T A089668 30487086144,223052123830,1544098243424,10208488021176,64917814932256,
%U A089668 399310478637476,2386386863086080,13906802738650816,79261768839946496,442921922267640894
%N A089668 a(n) = S2(n,5), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.
%H A089668 G. C. Greubel, <a href="/A089668/b089668.txt">Table of n, a(n) for n = 0..1000</a>
%H A089668 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 A089668 a(n) = (1/128)*n*(21*n^5 + 61*n^4 + 55*n^3 + 15*n^2 - 28*n + 4)*4^n - (1/48)*n^2*(n-1)^2*(3*n-5)*(n^2 + 4*n - 6)*binomial(2*n, n)/((2*n-1)*(2*n-3)). (See Wang and Zhang, p. 338)
%F A089668 From _G. C. Greubel_, May 25 2022: (Start)
%F A089668 a(n) = (1/2)*(n*(21*n^5 + 61*n^4 + 55*n^3 + 15*n^2 - 28*n + 4)*4^(n-3) - (n-1)*(3*n-5)*(n^2 + 4*n - 6)*binomial(n+1, 3)*Catalan(n-2)).
%F A089668 G.f.: x*( 4*(1 + 103*x + 1012*x^2 + 1688*x^3 + 512*x^4 - 256*x^5) - 3*x*(1 + 54*x + 26*x^2 - 156*x^3 - 104*x^4 + 320*x^5 -240*x^6)*sqrt(1-4*x) )/(1-4*x)^7. (End)
%t A089668 Table[(1/2)*(n*(21*n^5+61*n^4+55*n^3+15*n^2-28*n+4)*4^(n-3) -(n-1)*(3*n-5)*(n^2 + 4*n-6)*Binomial[n+1, 3]*CatalanNumber[n-2]), {n, 0, 40}] (* _G. C. Greubel_, May 25 2022 *)
%o A089668 (SageMath) [(1/2)*(n*(21*n^5 + 61*n^4 + 55*n^3 + 15*n^2 - 28*n + 4)*4^(n-3) - (n-1)*(3*n-5)*(n^2 + 4*n - 6)*binomial(n+1, 3)*catalan_number(n-2)) for n in (0..40)] # _G. C. Greubel_, May 25 2022
%Y A089668 Sequences of S2(n, t): A003583 (t=0), A089664 (t=1), A089665 (t=2), A089666 (t=3), A089667 (t=4), this sequence (t=5).
%Y A089668 Cf. A000108, A089658, A089669.
%K A089668 nonn,easy
%O A089668 0,2
%A A089668 _N. J. A. Sloane_, Jan 04 2004