OFFSET
0,8
COMMENTS
Column k is the diagonal of the rational function 1 / (1 - Sum_{j=1..k} x_j - Product_{j=1..k} x_j) for k>1. - Seiichi Manyama, Jul 10 2020
LINKS
Alois P. Heinz, Antidiagonals n = 0..44, flattened
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
EXAMPLE
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, ...
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 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 23 2013
STATUS
approved