OFFSET
0,3
COMMENTS
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
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)
MATHEMATICA
a[n_]:= n*Binomial[2*n-2, n-1]; Array[a, 30, 0] (* Amiram Eldar, Mar 10 2022 *)
PROG
(PARI) a(n) = n*binomial(2*n-2, n-1); \\ Joerg Arndt, Sep 04 2017
(Magma) [0] cat [n^2*Catalan(n-1): n in [1..30]]; // G. C. Greubel, Jun 19 2022
(SageMath) [n^2*catalan_number(n-1) for n in (0..30)] # G. C. Greubel, Jun 19 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Zerinvary Lajos, Oct 02 2007
STATUS
approved