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

0, 1, 68, 1314, 18080, 197350, 1836792, 15233316, 115776768, 821760390, 5520171800, 35438827996, 219038609088, 1310833221724, 7629754810160, 43348888067400, 241117582878720, 1316197491501510, 7065439665315480, 37362065079691500, 194909773207512000, 1004374157379474420

Offset

0,3

References

H. W. Gould, Combinatorial Identities, 1972. (See formulas 3.77, 3.78, and 3.79 on page 31.)

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

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)).

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

Mathematica

Table[Sum[m^6*(Binomial[n, m])^2, {m, 0, n}], {n, 21}]

Prog

(PARI) a(n) = sum(m=0, n, m^6*binomial(n, m)^2); \\ Jinyuan Wang, Nov 23 2019
(Magma) [(&+[Binomial(n, k)^2*k^6: k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 23 2022
(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

Crossrefs

Cf. A037966, A037972, A074334, A294486, A329521, A329913.

Sequence in context: A283575 A231060 A245875 * A230687 A337962 A231106

Adjacent sequences: A329441 A329442 A329443 * A329445 A329446 A329447

Keyword

nonn

Author

Nikita D. Gogin, Nov 16 2019

Status

approved