A007746 Number of ways for n-3 nonintersecting loops to cross a line 2n times.

42, 640, 5894, 42840, 271240, 1569984, 8536890, 44346456, 222516030, 1086685600, 5193298110, 24384586200, 112831907760, 515709552000, 2332549535400, 10455495457248, 46500885666900, 205401168733824, 901819865269180, 3938266773556720, 17116175702216624

Offset: 4

Author

Philippe Di Francesco (philippe(AT)amoco.saclay.cea.fr)

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 4..50
• P. Di Francesco, O. Golinelli and E. Guitter, Meanders: a direct enumeration approach, Nucl. Phys. B 482 [ FS ] (1996) 497-535.
• S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Theoretical Computer Science Vol. 117, pp. 227-241, 1993.

Crossrefs

A diagonal of triangle A008828.

Programs

Magma

[4*Factorial(2*n)/(3*Factorial(n-4)*Factorial(n+6))* (n^4+20*n^3+107*n^2-107*n+15): n in [4..25]]; // Vincenzo Librandi, Nov 23 2015

Mathematica

Table[4 (2 n)!/(3 (n - 4)! (n+6)!) (n^4 + 20 n^3 + 107 n^2 - 107 n + 15), {n, 4, 30}] (* Vincenzo Librandi, Nov 23 2015 *)

Formula

a(n) = 4 * (2*n)! * (n^4+20*n^3+107*n^2-107*n+15) / ( 3*(n-4)! * (n+6)! ).