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
Examples
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)].
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..25
Programs
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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 *) -
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
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