A153336 Number of zig-zag paths from top to bottom of a 2n by 2n square whose color is that of the top right corner.

1, 8, 52, 296, 1556, 7768, 37416, 175568, 807604, 3657464, 16357496, 72407728, 317777032, 1384524656, 5994736336, 25816193952, 110652549620, 472302724408, 2008499580504, 8513063608304, 35975584631128, 151621915797840

Offset: 1

Author

Joseph Myers, Dec 24 2008, Dec 31 2008

Keywords

• easy
• nonn

Examples

			a(3) = (2*3 + 1)*2 ^ (2*3 - 2) - 2*(2*3 - 1) * binomial(2*3 - 2, 3 - 1) = 52. - _Indranil Ghosh_, Feb 19 2017
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 1..1000
• Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation

Crossrefs

Cf. A102699, A153334, A153335, A153337, A153338.

Programs

Mathematica

Table[(2n+1) 2^(2n-2)-2(2n-1) Binomial[2n-2,n-1],{n,1,22}] (* Indranil Ghosh, Feb 19 2017 *)

PARI

a(n) = (2*n+1)*2^(2*n-2) - 2*(2*n-1)*binomial(2*n-2, n-1); \\ Michel Marcus, Feb 19 2017

Python

import math
def C(n,r):
    f=math.factorial
    return f(n)/f(r)/f(n-r)
def A153336(n):
    return str((2*n+1)*2**(2*n-2)-2*(2*n-1)*C(2*n-2,n-1)) # Indranil Ghosh, Feb 19 2017

Formula

a(n) = (2n+1)2^(2n-2) - 2(2n-1)binomial(2n-2,n-1).