A037972 a(n) = n^2*(n+1)*binomial(2*n-2, n-1)/2.

0, 1, 12, 108, 800, 5250, 31752, 181104, 988416, 5212350, 26741000, 134132856, 660284352, 3199016548, 15288882000, 72209880000, 337535723520, 1563410094390, 7182839945160, 32761238433000, 148450107960000, 668693511305820, 2995943329133040, 13356820221694560

Offset

0,3

References

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

Links

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

Formula

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

a(n) = A000217(n)*A037965(n). - R. J. Mathar, Jul 26 2015

(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

(n-1)^3*a(n) = 2*n*(n+1)*(2*n-3)*a(n-1). - R. J. Mathar, Oct 20 2015

G.f.: x * (1 - 2*x + 10*x^2 - 12*x^3) / (1 - 4*x)^(7/2). - Ilya Gutkovskiy, Nov 17 2021

Mathematica

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

Prog

(PARI) {a(n) = n^2*(n+1)*binomial(2*n-2, n-1)/2} \\ Seiichi Manyama, Aug 09 2020
(Magma) [n^2*(n+1)*Binomial(2*n-2, n-1)/2: n in [0..30]]; // G. C. Greubel, Jun 22 2022
(SageMath) [n^2*binomial(n+1, 2)*catalan_number(n-1) for n in (0..30)] # G. C. Greubel, Jun 22 2022

Crossrefs

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

Sequence in context: A321672 A241230 A353047 * A111990 A053469 A055533

Adjacent sequences: A037969 A037970 A037971 * A037973 A037974 A037975

Keyword

nonn

Author

N. J. A. Sloane

Status

approved