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A089667 a(n) = S2(n,4), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.

%I A089667 #25 Feb 01 2025 23:17:26
%S A089667 0,4,265,5984,85722,944904,8771462,72095520,541127988,3785356752,
%T A089667 25032083230,158102986624,961123994220,5656943319664,32386277835772,
%U A089667 181019819948864,990793669704552,5323620638111136,28137973407708174,146552649537716992
%N A089667 a(n) = S2(n,4), where S2(n, t) = Sum_{k=0..n} k^t *(Sum_{j=0..k} binomial(n,j))^2.
%H A089667 G. C. Greubel, <a href="/A089667/b089667.txt">Table of n, a(n) for n = 0..1000</a>
%H A089667 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 A089667 a(n) = (1/480)*( n*(n+1)*(93*n^3 + 132*n^2 + 53*n - 38)*4^n - 4*n*(n-1)*(15*n^5 - 99*n^3 + 116*n^2 - 34*n + 6)*binomial(2*n, n)/((2*n-1)*(2*n-3)) ). (See Wang and Zhang, p. 338.)
%F A089667 From _G. C. Greubel_, May 25 2022: (Start)
%F A089667 a(n) = (1/30)*( n*(n+1)*(93*n^3 + 132*n^2 + 53*n - 38)*4^(n-2) - (n-1)*(15*n^5 - 99*n^3 + 116*n^2 - 34*n + 6)*Catalan(n-2) ).
%F A089667 G.f.: x*( 4*(1 + 43*x + 160*x^2 + 96*x^3) - x*(3 + 62*x - 72*x^2 + 96*x^3 - 224*x^4 + 144*x^5)*sqrt(1-4*x) )/(1-4*x)^6. [Typo corrected by _Georg Fischer_, Nov 09 2022] (End)
%t A089667 Table[(1/30)*(n*(n+1)*(93*n^3+132*n^2+53*n-38)*4^(n-2) -(n-1)*(15*n^5-99*n^3 + 116*n^2-34*n+6)*CatalanNumber[n-2]), {n,0,40}] (* _G. C. Greubel_, May 25 2022 *)
%t A089667 CoefficientList[Series[x*( 4*(1 + 43*x + 160*x^2 + 96*x^3) - x*(3 + 62*x - 72*x^2 + 96*x^3 - 224*x^4 + 144*x^5)*Sqrt[1-4*x] )/(1-4*x)^6, {x,0,35}], x] (* _Georg Fischer_, Nov 09 2022 *)
%o A089667 (SageMath) [(1/30)*(n*(n+1)*(93*n^3+132*n^2+53*n-38)*4^(n-2) - (n-1)*(15*n^5 - 99*n^3+116*n^2-34*n+6)*catalan_number(n-2) ) for n in (0..40)] # _G. C. Greubel_, May 25 2022
%Y A089667 Sequences of S2(n, t): A003583 (t=0), A089664 (t=1), A089665 (t=2), A089666 (t=3), this sequence (t=4), A089668 (t=5).
%Y A089667 Cf. A000108, A089658, A089669.
%K A089667 nonn,easy
%O A089667 0,2
%A A089667 _N. J. A. Sloane_, Jan 04 2004