A229267 Number of lattice paths from {n}^n to {0}^n using steps that decrement one component or all components by 1.

1, 1, 13, 2371, 67982041, 629799991355641, 2672932604015450235911761, 7364217994146042440421602767480184881, 18165821273625565354157327818616137066973745155992321, 53130704578476340997304138835621075610747224340706918846011664495415681

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..25

Formula

a(n) = Sum_{j=0..n} multinomial(n+(n-1)*j; n-j, {j}^n) for n>1, a(0) = a(1) = 1.

Example

a(2) = 3*3 + 2*2 = 13:
.           (0,2)
.          /     \
.     (1,2)-------(0,1)
.    /     \     /     \
(2,2)-------(1,1)-------(0,0)
.    \     /     \     /
.     (2,1)-------(1,0)
.          \     /
.           (2,0)

Maple

with(combinat):
a:= n-> `if`(n<2, 1, add(multinomial(n+(n-1)*j, n-j, j$n), j=0..n)):
seq(a(n), n=0..10);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!); a[n_] := If[n < 2, 1, Sum[multinomial[n+(n-1)*j, Join[{n-j}, Array[j&, n]]], {j, 0, n}]]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

Crossrefs

Main diagonal of A229142.

Sequence in context: A221901 A096721 A301466 * A350308 A283099 A348817

Adjacent sequences: A229264 A229265 A229266 * A229268 A229269 A229270

Keyword

nonn

Author

Alois P. Heinz, Sep 23 2013

Status

approved