OFFSET
0,8
COMMENTS
A(n,k) is also the number of binary words with n*k 1's and n 0's such that for every prefix the number of 1's is >= the number of 0's. The A(2,2) = 9 words are: 101011, 101101, 101110, 110011, 110101, 110110, 111001, 111010, 111100.
LINKS
Alois P. Heinz, Antidiagonals n = 0..140
Paul Barry, On the Central Antecedents of Integer (and Other) Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.3.
Wikipedia, Young tableau
FORMULA
A(n,k) = max(0, C((k+1)*n,n)*((k-1)*n+1)/(k*n+1)).
From Peter Bala, Aug 17 2026: (Start)
For n, k >= 1, A(n, k) = (k + 1)*binomial((k+1)*n-1, n-1) - binomial((k+1)*n, n-1) (clearly integral).
For k >= 1, A(n, k) = the constant term in the Laurent expansion of ( (1 + T(x))^(k+1)/T(x) )^n, where T(x) = Sum_{n >= 1} n^(n-1)*x^n/n! is the tree function - see A000169. (End)
EXAMPLE
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 2, 9, 20, 35, 54, 77, ...
0, 5, 48, 154, 350, 663, 1120, ...
0, 14, 275, 1260, 3705, 8602, 17199, ...
0, 42, 1638, 10659, 40480, 115101, 272272, ...
MAPLE
A:= (n, k)-> max(0, binomial((k+1)*n, n)*((k-1)*n+1)/(k*n+1)):
seq(seq(A(n, d-n), n=0..d), d=0..12);
MATHEMATICA
a[n_, k_] := Max[0, Binomial[(k+1)*n, n]*((k-1)*n+1)/(k*n+1)]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Oct 01 2013, after Maple *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 28 2012
STATUS
approved