OFFSET
0,6
COMMENTS
A(n,k) is the number of set partitions of [n] into blocks of size > k.
LINKS
Seiichi Manyama, Antidiagonals n = 0..139, flattened
E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73.
FORMULA
E.g.f. of column k: Product_{i>k} exp(x^i/i!).
A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = Sum_{i=k..n-1} binomial(n-1,i)*A(n-1-i,k) for n > k.
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 0, 0, 0, 0, 0, 0, 0, ...
2, 1, 0, 0, 0, 0, 0, 0, ...
5, 1, 1, 0, 0, 0, 0, 0, ...
15, 4, 1, 1, 0, 0, 0, 0, ...
52, 11, 1, 1, 1, 0, 0, 0, ...
203, 41, 11, 1, 1, 1, 0, 0, ...
877, 162, 36, 1, 1, 1, 1, 0, ...
MAPLE
A:= proc(n, k) option remember; `if`(n=0, 1, add(
A(n-j, k)*binomial(n-1, j-1), j=1+k..n))
end:
seq(seq(A(n, d-n), n=0..d), d=0..14); # Alois P. Heinz, Sep 28 2017
MATHEMATICA
A[0, _] = 1;
A[n_, k_] /; 0 <= k <= n := A[n, k] = Sum[A[n-j, k] Binomial[n-1, j-1], {j, k+1, n}];
A[_, _] = 0;
Table[A[n-k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 06 2019 *)
PROG
(Ruby)
def ncr(n, r)
return 1 if r == 0
(n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
ary = [1]
(1..n).each{|i| ary << (k..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ary[-1 - j]}}
ary
end
def A293024(n)
a = []
(0..n).each{|i| a << A(i, n - i)}
ary = []
(0..n).each{|i|
(0..i).each{|j|
ary << a[i - j][j]
}
}
ary
end
p A293024(20)
CROSSREFS
Main diagonal gives A000007.
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Sep 28 2017
STATUS
approved