OFFSET
0,3
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.
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
MATHEMATICA
Array[#^2*Binomial[2#-2, #-1] &, 27, 0] (* Michael De Vlieger, Jul 15 2020 *)
PROG
(PARI) {a(n) = n^2*binomial(2*n-2, n-1)} \\ Seiichi Manyama, Jul 15 2020
(Magma) [0] cat [n^3*Catalan(n-1): n in [1..30]]; // G. C. Greubel, Jun 19 2022
(SageMath) [n^3*catalan_number(n-1) for n in (0..30)] # G. C. Greubel, Jun 19 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Seiichi Manyama, Jul 15 2020
STATUS
approved