|
| |
|
|
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)}.
|
|
5
|
|
|
|
1, 1, 4, 20, 126, 882, 6732, 54483, 461890, 4059770, 36749648, 340841228, 3226474132, 31079221500, 303907314960, 3010947401340, 30176604541890, 305544118531410, 3122029001183400, 32162924697555960, 333798776316127980, 3487606641543204180, 36662847052669011720, 387575171486963664750
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
LINKS
|
|
|
|
FORMULA
|
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
|
|
|
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 *)
|
|
|
PROG
|
(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))};
(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
(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
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,walk
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
