A229142 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component or all components by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.
1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 13, 1, 1, 1, 25, 115, 63, 1, 1, 1, 121, 2641, 2371, 321, 1, 1, 1, 721, 114121, 392641, 54091, 1683, 1, 1, 1, 5041, 7489441, 169417921, 67982041, 1307377, 8989, 1, 1, 1, 40321, 681120721, 137322405361, 308238414121, 12838867105, 32803219, 48639, 1, 1
Offset: 0
Examples
A(1,3) = 3*2+1 = 7:
(0,1,1)-(0,0,1)
/ X \
(1,1,1)-(1,0,1) (0,1,0)-(0,0,0)
\ \ X / /
\ (1,1,0)-(1,0,0) /
`---------------´
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 3, 7, 25, 121, ...
1, 1, 13, 115, 2641, 114121, ...
1, 1, 63, 2371, 392641, 169417921, ...
1, 1, 321, 54091, 67982041, 308238414121, ...
1, 1, 1683, 1307377, 12838867105, 629799991355641, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..44, flattened
Crossrefs
Programs
-
Maple
with(combinat): A:= (n,k)-> `if`(k<2, 1, add(multinomial(n+(k-1)*j, n-j, j$k), j=0..n)): seq(seq(A(n, d-n), n=0..d), d=0..10);
-
Mathematica
a[, 0] = a[, 1] = 1; a[n_, k_] := Sum[Product[Binomial[n+j*m, m], {j, 0, k-1}], {m, 0, n}]; Table[a[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 11 2013 *)
Formula
A(n,k) = Sum_{j=0..n} multinomial(n+(k-1)*j; n-j, {j}^k) for k>1, A(n,0) = A(n,1) = 1.
G.f. of column k: Sum_{j>=0} (k*j)!/j!^k * x^j / (1-x)^(k*j+1). for k>1. - Seiichi Manyama, Jul 10 2020
Comments