OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..550
G. Kreweras and H. Niederhausen, Solution of an enumerative problem connected with lattice paths, European J. Combin., 2 (1981), 55-60.
FORMULA
G.f.: expression with a 2F1 function and an anti-derivative, see Maple program below. - Mark van Hoeij, May 01 2013
a(n) ~ 2^(6*n + 1/2) / (Pi * n^3). - Vaclav Kotesovec, Dec 16 2017
D-finite with recurrence (2*n+1)*(n+1)^2*a(n) -4*(4*n+1)*(2*n-1)*(4*n-1)*a(n-1)=0. - R. J. Mathar, Feb 08 2021
From G. C. Greubel, Feb 12 2021: (Start)
a(n) = binomial(4*n+1, 2*n+1)*binomial(2*n, n)/(n+1)^2.
a(n) = ((4*n+1)/(n+1))*C_{n}*C_{2*n}, where C_{n} are the Catalan numbers (A000108). (End)
MAPLE
a:=n->1/(2*n+1)!*(4*n+1)!/(n+1)!^2: seq(a(n), n=1..17);
ogf := -1/(4*x)-Int(x^(-3/2)*hypergeom([-1/4, 1/4], [1], 64*x), x)/(8*x^(1/2));
series( eval(ogf, Int = proc(a, x) int(series(a, x=0, 32), x) end), x=0, 30); # Mark van Hoeij, May 01 2013
MATHEMATICA
Table[((4*n+1)/(n+1))*CatalanNumber[n]*CatalanNumber[2*n], {n, 0, 30}] (* G. C. Greubel, Feb 12 2021 *)
PROG
(Sage) [((4*n+1)/(n+1))*catalan_number(n)*catalan_number(2*n) for n in (0..30)] # G. C. Greubel, Feb 12 2021
(Magma) [((4*n+1)/(n+1))*Catalan(n)*Catalan(2*n): n in [0..30]]; // G. C. Greubel, Feb 12 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Aug 19 2005
STATUS
approved