A111911 a(n) = (4*n+1)!/( (2*n+1)! * ((n+1)!)^2 ).

1, 5, 84, 2145, 68068, 2469012, 98062800, 4159088505, 185392049700, 8592433629780, 410935420867920, 20167102448028900, 1011343194858833424, 51656474975499371600, 2680436673901084633920, 141007991981718802584105, 7507710828193055843153700

Offset

0,2

Comments

Main diagonal of the square array A111910, i.e., a(n) = A111910(n,n).

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

Cf. A111910.

Cf. A000108.

Sequence in context: A258391 A061628 A193369 * A156720 A288163 A268880

Adjacent sequences: A111908 A111909 A111910 * A111912 A111913 A111914

Keyword

nonn

Author

Emeric Deutsch, Aug 19 2005

Status

approved