A262810 Number of lattice paths from {n}^n to {0}^n using steps that decrement one or more components by one.

1, 1, 13, 16081, 5552351121, 1050740615666453461, 179349571255187154941191217629, 41020870889694863957061607086939138327565057, 17469051230066445323872793284679234619523576313653708533767425

Offset

0,3

Links

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

Formula

a(n) = A262809(n,n).

a(n) ~ n^(n^2 - n/2 + 1) / (exp(1/12) * 2^(n + log(2)/24) * Pi^((n-1)/2) * log(2)^(n^2+1)). - Vaclav Kotesovec, Mar 23 2016

Example

a(2) = 13: [(2,2),(1,2),(0,2),(0,1),(0,0)], [(2,2),(1,2),(0,1),(0,0)], [(2,2),(1,2),(1,1),(0,1),(0,0)], [(2,2),(1,2),(1,1),(0,0)], [(2,2),(1,2),(1,1),(1,0),(0,0)], [(2,2),(2,1),(1,1),(0,1),(0,0)], [(2,2),(2,1),(1,1),(0,0)], [(2,2),(2,1),(1,1),(1,0),(0,0)], [(2,2),(2,1),(2,0),(0,1),(0,0)], [(2,2),(2,1),(1,0),(0,0)], [(2,2),(1,1),(0,1),(0,0)], [(2,2),(1,1),(0,0)], [(2,2),(1,1),(1,0),(0,0)].

Mathematica

Flatten[{1, Table[Sum[Sum[(-1)^i*Binomial[j, i]*Binomial[j - i, n]^n, {i, 0, j}], {j, 0, n^2}], {n, 1, 10}]}] (* Vaclav Kotesovec, Mar 23 2016 *)

Prog

(PARI) a(n)=sum(j=0, n^2, sum(i=0, j, (-1)^i*binomial(j, i)*binomial(j - i, n)^n)) \\ Charles R Greathouse IV, Jul 29 2016

Crossrefs

Main diagonal of A262809.

Cf. A316677.

Sequence in context: A176096 A215685 A327585 * A340295 A203514 A098562

Adjacent sequences: A262807 A262808 A262809 * A262811 A262812 A262813

Keyword

nonn,nice

Author

Alois P. Heinz, Oct 02 2015

Status

approved