A037972 - OEIS

%I #37 Jun 22 2022 07:54:10

%S 0,1,12,108,800,5250,31752,181104,988416,5212350,26741000,134132856,

%T 660284352,3199016548,15288882000,72209880000,337535723520,

%U 1563410094390,7182839945160,32761238433000,148450107960000,668693511305820,2995943329133040,13356820221694560

%N a(n) = n^2*(n+1)*binomial(2*n-2, n-1)/2.

%D Identity (3.78), S_{3}, in H. W. Gould, Combinatorial Identities, Morgantown, 1972, page 31.

%H Seiichi Manyama, <a href="/A037972/b037972.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..n} k^3*(C(n,k))^2. [heruneedollar (heruneedollar(AT)gmail.com), Mar 20 2010]

%F a(n) = A000217(n)*A037965(n). - _R. J. Mathar_, Jul 26 2015

%F (n-1)^2*a(n) = 2*(11*n-16)*a(n-1) + 8*n*(2*n-5)*a(n-2). - _R. J. Mathar_, Oct 20 2015

%F (n-1)^3*a(n) = 2*n*(n+1)*(2*n-3)*a(n-1). - _R. J. Mathar_, Oct 20 2015

%F G.f.: x * (1 - 2*x + 10*x^2 - 12*x^3) / (1 - 4*x)^(7/2). - _Ilya Gutkovskiy_, Nov 17 2021

%t Table[n^2*Binomial[n+1,2]*CatalanNumber[n-1], {n,0,30}] (* _G. C. Greubel_, Jun 22 2022 *)

%o (PARI) {a(n) = n^2*(n+1)*binomial(2*n-2, n-1)/2} \\ _Seiichi Manyama_, Aug 09 2020

%o (Magma) [n^2*(n+1)*Binomial(2*n-2, n-1)/2: n in [0..30]]; // _G. C. Greubel_, Jun 22 2022

%o (SageMath) [n^2*binomial(n+1,2)*catalan_number(n-1) for n in (0..30)] # _G. C. Greubel_, Jun 22 2022

%Y Cf. A000108, A000217, A037965, A037966, A074334, A329913, A329444.

%K nonn

%O 0,3

%A _N. J. A. Sloane_