A151341 Number of walks within N^2 (the first quadrant of Z^2) starting and ending at (0,0) and consisting of 2 n steps taken from {(-1, -1), (-1, 0), (-1, 1), (1, 0)}.

1, 1, 4, 20, 126, 882, 6732, 54483, 461890, 4059770, 36749648, 340841228, 3226474132, 31079221500, 303907314960, 3010947401340, 30176604541890, 305544118531410, 3122029001183400, 32162924697555960, 333798776316127980, 3487606641543204180, 36662847052669011720, 387575171486963664750

Offset: 0

Author

Manuel Kauers, Nov 18 2008

Keywords

• nonn
• walk

Links

• G. C. Greubel, Table of n, a(n) for n = 0..925
• M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.

Crossrefs

Cf. A000108, A001006.

Programs

Magma

[(3/2)^(n+2)*Catalan(n)*(&+[Binomial(k, n-k+2)*Catalan(k-1)/(3^k): k in [1..n+2]]): n in [0..25]]; // G. C. Greubel, Mar 11 2019

Maple

ogf := subs(t=sqrt(x),series( Int(Int(2*hypergeom([3/2, 3/2],[3],16*t^2/(1+4*t^2))/(1+4*t^2)^(3/2),t),t)/t^2, t=0, 60)); # Mark van Hoeij, Aug 17 2014

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[1 + i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[aux[0, 0, 2 n], {n, 0, 25}]
Table[CatalanNumber[n]*(3/2)^(n+2)*Sum[CatalanNumber[k-1]*Binomial[k, n+2 -k]/3^k, {k, 1, n+2}], {n, 0, 25}] (* G. C. Greubel, Mar 11 2019 *)

PARI

{a(n) = (3/2)^(n+2)*(binomial(2*n, n)/(n+1))*sum(k=1, n+2, binomial(k, n-k+2)*binomial(2*k-2, k-1)/(3^k*k))};
vector(25, n, n--; a(n)) \\ G. C. Greubel, Mar 11 2019

Sage

[(3/2)^(n+2)*catalan_number(n)*sum(binomial(k, n-k+2)* catalan_number(k-1)/3^k for k in (1..n+2)) for n in (0..25)] # G. C. Greubel, Mar 11 2019

Formula

a(n) = A000108(n)*A001006(n).

Conjecture: n*(n+2)*(n+1)*a(n) - 2*n*(2*n-1)*(2*n+1)*a(n-1) -12*(n-1)*(2*n-1)*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Jul 21 2017