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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. 14
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 (list; table; graph; refs; listen; history; text; internal format)
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
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
Rows n=0-1 give: A000012, A038507 (for k>1).
Main diagonal gives: A229267.
Sequence in context: A360289 A343717 A263159 * A156535 A365427 A366787
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 23 2013
STATUS
approved

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Last modified July 21 22:43 EDT 2024. Contains 374478 sequences. (Running on oeis4.)