A037966 a(n) = n^2*binomial(2*n-2, n-1).

0, 1, 8, 54, 320, 1750, 9072, 45276, 219648, 1042470, 4862000, 22355476, 101582208, 457002364, 2038517600, 9026235000, 39710085120, 173712232710, 756088415280, 3276123843300, 14138105520000, 60790319209620, 260516811228960, 1113068351807880, 4742456099097600, 20154752301937500, 85453569951920352

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1972.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• Nikita Gogin and Mika Hirvensalo, On the Moments of Squared Binomial Coefficients, (2020).
• Han Mao Kiah, Alexander Vardy, and Hanwen Yao, Efficient Algorithms for the Bee-Identification Problem, arXiv:2212.09952 [cs.IT], 2022.

Crossrefs

Cf. A000108, A000984, A037965, A336214.

Programs

Magma

[0] cat [n^3*Catalan(n-1): n in [1..30]]; // G. C. Greubel, Jun 19 2022

Mathematica

Array[#^2*Binomial[2#-2, #-1] &, 27, 0] (* Michael De Vlieger, Jul 15 2020 *)

PARI

{a(n) = n^2*binomial(2*n-2, n-1)} \\ Seiichi Manyama, Jul 15 2020

SageMath

[n^3*catalan_number(n-1) for n in (0..30)] # G. C. Greubel, Jun 19 2022

Formula

a(n) = Sum_{k=0..n} k^2*binomial(n,k)^2. - Paul Barry, Mar 04 2003

a(n) = n^2*A000984(n-1). - Zerinvary Lajos, Jan 18 2007, corrected Jul 26 2015

a(n) = n*A037965(n). - Zerinvary Lajos, Jan 18 2007, corrected Jul 26 2015

(n-1)^3*a(n) = 2*n^2*(2*n-3)*a(n-1). - R. J. Mathar, Jul 26 2015

E.g.f.: x*exp(2*x)*((1 + 2*x)*BesselI(0,2*x) + 2*x*BesselI(1,2*x)). - Ilya Gutkovskiy, Mar 04 2021

Extensions

More terms from Seiichi Manyama, Jul 15 2020