A037965 a(n) = n*binomial(2*n-2, n-1).

0, 1, 4, 18, 80, 350, 1512, 6468, 27456, 115830, 486200, 2032316, 8465184, 35154028, 145608400, 601749000, 2481880320, 10218366630, 42004911960, 172427570700, 706905276000, 2894777105220, 11841673237680, 48394276165560, 197602337462400, 806190092077500

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n+1) is the convolution of A000984 and A081294. - Paul Barry, Sep 18 2008

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000984, A001622, A081294, A109188 (inverse binomial transform).

Cf. A000108, A100071, A134757.

Programs

Magma

[0] cat [n^2*Catalan(n-1): n in [1..30]]; // G. C. Greubel, Jun 19 2022

Mathematica

a[n_]:= n*Binomial[2*n-2, n-1]; Array[a, 30, 0] (* Amiram Eldar, Mar 10 2022 *)

PARI

a(n) = n*binomial(2*n-2, n-1); \\ Joerg Arndt, Sep 04 2017

SageMath

[n^2*catalan_number(n-1) for n in (0..30)] # G. C. Greubel, Jun 19 2022

Formula

Assuming offset -1 here and offset 0 in A134757, A134757 is the inverse binomial transform of this sequence. - Gary W. Adamson, Nov 08 2007

G.f.: Hypergeometric2F1([1/2, 2], [1], 4*x). - Paul Barry, Sep 03 2008

From Paul Barry, Sep 18 2008: (Start)

G.f.: x*(1-2*x)/(1-4*x)^(3/2);

a(n+1) = Sum_{k=0..n} binomial(2*k,k)*(4^(n-k) + 0^(n-k))/2. (End)

D-finite with recurrence (n-1)*a(n) - 2*(3*n-4)*a(n-1) + 4*(2*n-5)*a(n-2) = 0. - R. J. Mathar, Nov 30 2012

E.g.f.: x*exp(2*x)*BesselI(0,2*x). - Ilya Gutkovskiy, Aug 22 2018

a(n) = n*A000984(n-1). - R. J. Mathar, Nov 08 2021

From Amiram Eldar, Mar 10 2022: (Start)

Sum_{n>=1} 1/a(n) = 4*Pi/(3*sqrt(3)) - Pi^2/9.

Sum_{n>=1} (-1)^(n+1)/a(n) = 8*log(phi)/sqrt(5) - 4*log(phi)^2, where phi is the golden ratio (A001622). (End)

Extensions

More terms from Zerinvary Lajos, Oct 02 2007