OFFSET
0,2
COMMENTS
This sequence is one half of the odd part of the bisection of A241530. The even part is given in A002894. - Wolfdieter Lang, Sep 06 2016
REFERENCES
E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 96.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..180
Ling Gao, Graph assembly for spider and tadpole graphs, Master's Thesis, Cal. State Poly. Univ. (2023). See p. 36.
Pedro J. Miana and Natalia Romero, Moments of combinatorial and Catalan numbers, Journal of Number Theory, Volume 130, Issue 8, August 2010, Pages 1876-1887. See Omega1 Remark 3 p. 1882.
Yidong Sun and Fei Ma, Four transformations on the Catalan triangle, arXiv preprint arXiv:1305.2017 [math.CO], 2013 (see Omega_1).
Yidong Sun and Fei Ma, Some new binomial sums related to the Catalan triangle, Electronic Journal of Combinatorics 21(1) (2014), #P1.33.
FORMULA
a(n) = C(2*n+1,n)*C(2*n,n) = A001700(n)*A000984(n) = A000984(n)*A000984(n+1)/2, n>=0. - Zerinvary Lajos, Jan 23 2007
G.f.: (EllipticK(4*x^(1/2)) - EllipticE(4*x^(1/2)))/(4*x*Pi). - Mark van Hoeij, Oct 24 2011
n*(n+1)*a(n) -4*(2*n-1)*(2*n+1)*a(n-1)=0. - R. J. Mathar, Sep 08 2013
0 = a(n)*(+65536*a(n+2) - 23040*a(n+3) + 1400*a(n+4)) + a(n+1)*(-1536*a(n+2) + 1184*a(n+3) - 90*a(n+4)) + a(n+2)*(-24*a(n+2) - 6*a(n+3) + a(n+4)) for all n in Z. - Michael Somos, May 28 2014
0 = a(n+1)^3 * (+256*a(n) - 6*a(n+1) + a(n+2)) + a(n) * a(n+1) * a(n+
2) * (-768*a(n) - 20*a(n+1) - 3*a(n+2)) + 90*a(n)^2*a(n+2)^2 for all n in Z. - Michael Somos, Sep 17 2014
a(n) = A241530(2n+1)/2, n >= 0. - Wolfdieter Lang, Sep 06 2016
a(n) ~ 2^(4*n+1)/(Pi*n). - Ilya Gutkovskiy, Sep 06 2016
EXAMPLE
G.f. = 1 + 6*x + 60*x^2 + 700*x^3 + 8820*x^4 + 116424*x^5 + ...
MAPLE
seq(binomial(2*n+1, n)*binomial(2*n, n), n=0..16); # Zerinvary Lajos, Jan 23 2007
MATHEMATICA
a[ n_] := Binomial[2 n + 1, n] Binomial[2 n, n]; (* Michael Somos, May 28 2014 *)
a[ n_] := SeriesCoefficient[ (EllipticK[ 16 x] - EllipticE[ 16 x]) / (4 x Pi), {x, 0, n}]; (* Michael Somos, May 28 2014 *)
Table[(2 n)!*(2 n + 1)!/((n + 1)!*n!^3), {n, 0, 16}] (* Michael De Vlieger, Sep 06 2016 *)
PROG
(Magma) [Factorial(2*n)*Factorial(2*n+1) /(Factorial(n+1)* Factorial(n)^3): n in [0..20]]; // Vincenzo Librandi, Oct 25 2011
(Haskell)
a000894 n = a132813 (2 * n) n -- Reinhard Zumkeller, Apr 04 2014
(PARI) {a(n) = binomial( 2*n + 1, n) * binomial( 2*n, n)}; /* Michael Somos, May 28 2014 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved