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A329444 The sixth moments of the squared binomial coefficients; a(n) = Sum_{m=0..n} m^6*binomial(n, m)^2.

%I A329444 #40 Jun 24 2022 09:30:19
%S A329444 0,1,68,1314,18080,197350,1836792,15233316,115776768,821760390,
%T A329444 5520171800,35438827996,219038609088,1310833221724,7629754810160,
%U A329444 43348888067400,241117582878720,1316197491501510,7065439665315480,37362065079691500,194909773207512000,1004374157379474420
%N A329444 The sixth moments of the squared binomial coefficients; a(n) = Sum_{m=0..n} m^6*binomial(n, m)^2.
%D A329444 H. W. Gould, Combinatorial Identities, 1972. (See formulas 3.77, 3.78, and 3.79 on page 31.)
%H A329444 Seiichi Manyama, <a href="/A329444/b329444.txt">Table of n, a(n) for n = 0..1000</a>
%F A329444 a(n) = binomial(2*n, n) * n^3*(n^6 + 3*n^5 - 13*n^4 - 15*n^3 + 30*n^2 + 8*n - 2)/(8*(2*n-1)*(2*n-3)*(2*n-5)).
%F A329444 G.f.: x*(1 + 42*x - 168*x^2 + 1648*x^3 - 7608*x^4 + 18144*x^5 - 19376*x^6 - 1440*x^7 + 14400*x^8)/((1-4*x)^6*sqrt(1-4*x)). - _G. C. Greubel_, Jun 23 2022
%t A329444 Table[Sum[m^6*(Binomial[n, m])^2, {m, 0, n}], {n, 21}]
%o A329444 (PARI) a(n) = sum(m=0, n, m^6*binomial(n, m)^2); \\ _Jinyuan Wang_, Nov 23 2019
%o A329444 (Magma) [(&+[Binomial(n,k)^2*k^6: k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Jun 23 2022
%o A329444 (SageMath) [n^3*(n+1)*(n^6+3*n^5-13*n^4-15*n^3+30*n^2+8*n-2)*catalan_number(n)/(8*(2*n-1)*(2*n-3)*(2*n-5)) for n in (0..30)] # _G. C. Greubel_, Jun 23 2022
%Y A329444 Cf. A037966, A037972, A074334, A294486, A329521, A329913.
%K A329444 nonn
%O A329444 0,3
%A A329444 _Nikita D. Gogin_, Nov 16 2019