0,2

Also equal to the number of standard tableaux of 2n cells with height less than or equal to 4. A005817(2n) - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Feb 22 2007

Also equal to Sum binomial(2n,2i)C(i)C(n-i) = (4/pi^2) Integral_{0 .. pi} Integral_{0^pi} (2cos(x)+2cos(y))^{2n} sin^2(x)sin^2(y) dx dy, since this counts walks of 2n steps in the nonnegative quadrant of an integer lattice that return to the origin (cf. R. K. Guy link below). - Andrew V. Sutherland (drew(AT)math.mit.edu), Nov 29 2007

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Olivier Bernardi, Bijective Counting of Tree-Rooted Maps and Shuffles of Parenthesis Systems, Electronic Journal of Combinatorics, Vol. 14 (2007), R9.

R. Cori et al., Shuffle of parenthesis systems and Baxter permutations, J. Combin. Theory, A 43 (1986), 1-22.

D. Gouyou-Beauchamps, Chemins sous-diagonaux et tableau de Young, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.

D. Gouyou-Beauchamps, Standard Young tableaux of height 4 and 5, Europ. J. Combinatorics, 10, 1989, 69-82.

Kiran S. Kedlaya and Andrew V. Sutherland, "Hyperelliptic curves, L-polynomials, and random matrices", preprint, 2008.

R. C. Mullin, On the average activity of a spanning tree of a rooted map, J. Combin. Theory, 3 (1967), 103-121.

Liviu I. Nicolaescu, Counting Morse functions on the 2-sphere, arXiv:math/0512496.

R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6

a(n)=binomial(2n, n)*binomial(2n+2, n+1)/[(n+1)(n+2)] = 2(2n+1)binomial(2n, n)^2/[(n+2)(n+1)^2].

(n+1)*n*a(n) = 4*(2n-1)*(2n-3)*a(n-1).

G.f. in Maple notation: (1/2)/x+1/768/(x^2*Pi)*((32-512*x)*EllipticK(4*x^(1/2))+(-32-512*x)*EllipticE(4*x^(1/2))); from Karol A. Penson (penson(AT)lptl.jussieu.fr), Oct 24 2003.

E.g.f.: 1/3*(8*x^2*BesselI(0, 2*x)^2-4*BesselI(0, 2*x)*BesselI(1, 2*x)*x-BesselI(1, 2*x)^2-8*BesselI(1, 2*x)^2*x^2)/x. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 29 2003

E.g.f. Sum_{n>=0} a(n)*x^(2n)/(2n)! = BesselI(1, 2x)^2/x^2 . - Michael Somos Jun 22 2005

c:=n->binomial(2*n, n)/(n+1): seq(c(n)*c(n+1), n=0..21); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 05 2007

(PARI) (alias(C, binomial)); a(n)=(C(2*n, n)-C(2*n, n-1))*(C(2*n+2, n+1)-C(2*n+2, n)) /* Michael Somos Jun 22 2005 */

Cf. A000108.

Cf. A005817.

Sequence in context: A051575 A121201 A051405 this_sequence A036075 A123881 A089845

Adjacent sequences: A005565 A005566 A005567 this_sequence A005569 A005570 A005571

nonn,easy

R. K. Guy, Simon Plouffe, njas

More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 20 2004