A151368 Number of walks within N^2 (the first quadrant of Z^2) starting and ending at (0,0) and consisting of n steps taken from {(-1, -1), (-1, 0), (-1, 1), (0, -1), (0, 1), (1, 0)}.

1, 0, 2, 3, 12, 40, 145, 560, 2240, 9156, 38724, 166320, 728508, 3239808, 14595438, 66543477, 306511920, 1424916064, 6679435048, 31544500416, 149986398848, 717562911000, 3452381033556, 16696661334496, 81136327037620, 396022179418240, 1940898351416600, 9548613568549380, 47143311987432240

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..1366

M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.

Formula

G.f.: Int(Int(2*hypergeom([3/4, 5/4],[2],64*x^3*(2*x+1)/(8*x^2-1)^2)/(1-8*x^2)^(3/2),x),x)/x^2. - Mark van Hoeij, Aug 17 2014

1024*(n+4)*(n+3)*(n+2)*(n+1)*a(n)+512*(5*n+23)*(n+4)*(n+3)*(n+2)*a(n+1)+32*(n+4)*(n+3)*(84*n^2+868*n+2243)*a(n+2)+32*(n+5)*(n+4)*(46*n^2+554*n+1671)*a(n+3)+4*(n+6)*(n+5)*(100*n^2+1372*n+4737)*a(n+4)+24*(n+6)*(n+7)*(n^2+17*n+75)*a(n+5)-2*(7*n+65)*(n+8)*(n+7)*(n+6)*a(n+6)-3*(n+10)*(n+9)*(n+8)*(n+7)*a(n+7)=0. - Robert Israel, Sep 03 2018

Maple

f:= gfun:-rectoproc({1024*(n+4)*(n+3)*(n+2)*(n+1)*a(n)+512*(5*n+23)*(n+4)*(n+3)*(n+2)*a(n+1)+32*(n+4)*(n+3)*(84*n^2+868*n+2243)*a(n+2)+32*(n+5)*(n+4)*(46*n^2+554*n+1671)*a(n+3)+4*(n+6)*(n+5)*(100*n^2+1372*n+4737)*a(n+4)+24*(n+6)*(n+7)*(n^2+17*n+75)*a(n+5)-2*(7*n+65)*(n+8)*(n+7)*(n+6)*a(n+6)-3*(n+10)*(n+9)*(n+8)*(n+7)*a(n+7),
a(0)=1, a(1)=0, a(2)=2, a(3)=3, a(4)=12, a(5)=40, a(6)=145}, a(n), remember):
map(f, [$0..100]); # Robert Israel, Sep 03 2018

Mathematica

aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, j, -1 + n] + aux[i, -1 + j, -1 + n] + aux[i, 1 + j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[aux[0, 0, n], {n, 0, 25}]

Crossrefs

Sequence in context: A012510 A012310 A082526 * A087650 A177699 A256881

Adjacent sequences: A151365 A151366 A151367 * A151369 A151370 A151371

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved