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%I #14 Oct 23 2023 11:16:55
%S 0,4,41,306,1966,11540,63726,336700,1720364,8562024,41718190,
%T 199753004,942561636,4392660376,20253510956,92519626200,419201709976,
%U 1885719209936,8428262686254,37453751742604,165575219275700,728534225415864,3191850894862564
%N a(n) = S2(n,1), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.
%H G. C. Greubel, <a href="/A089664/b089664.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) = (1/8)*(n*(3*n+5)*4^n - 2*n*(n-1)*binomial(2*n, n)). (see Wang and Zhang, p. 338)
%F From _G. C. Greubel_, May 25 2022: (Start)
%F a(n) = (1/2)*( n*(3*n+5)*4^(n-1) - 3*binomial(n+1, 3)*Catalan(n) ).
%F G.f.: x*(4*(1-x) - 3*x*sqrt(1-4*x))/(1-4*x)^3.
%F E.g.f.: 2*x*(2 + 3*x)*exp(4*x) - (x^2/2)*(3*BesselI(0, 2*x) + 4*BesselI(1, 2*x) + BesselI(2, 2*x))*exp(2*x)). (End)
%t Table[(n*(3*n+5)*4^n -2*n*(n-1)*Binomial[2*n,n])/8, {n,0,40}] (* _G. C. Greubel_, May 25 2022 *)
%o (SageMath) [(1/2)*(n*(3*n+5)*4^(n-1) -3*binomial(n+1, 3)*catalan_number(n)) for n in (0..40)] # _G. C. Greubel_, May 25 2022
%o (PARI) a(n)=n*(3*n+5)*2^(2*n-3) - 3*binomial(n+1,3)*binomial(2*n,n)/(n+1)/2 \\ _Charles R Greathouse IV_, Oct 23 2023
%Y Sequences of S2(n, t): A003583 (t=0), this sequence (t=1), A089665 (t=2), A089666 (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
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