A144303 Square array A(n,m), n>=0, m>=0, read by antidiagonals: A(n,m) = n-th number of the m-th iteration of the hyperbinomial transform on the sequence of 1's.

1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 13, 29, 1, 1, 5, 22, 81, 212, 1, 1, 6, 33, 163, 689, 2117, 1, 1, 7, 46, 281, 1564, 7553, 26830, 1, 1, 8, 61, 441, 2993, 18679, 101961, 412015, 1, 1, 9, 78, 649, 5156, 38705, 268714, 1639529, 7433032, 1, 1, 10, 97, 911, 8257, 71801, 592489, 4538209, 30640257, 154076201, 1

Offset: 0

Author

Alois P. Heinz, Sep 17 2008, revised Oct 30 2012

Keywords

• nonn
• tabl

Comments

See A088956 for the definition of the hyperbinomial transform.

A(n,m), n>=0, m>=0, is the number of subtrees of the complete graph K_{n+m} on n+m vertices containing a given, fixed subtree on m vertices. - Alex Chin, Jul 25 2013

Examples

			Square array begins:
  1,     1,      1,      1,      1,       1,       1, ...
  1,     2,      3,      4,      5,       6,       7, ...
  1,     6,     13,     22,     33,      46,      61, ...
  1,    29,     81,    163,    281,     441,     649, ...
  1,   212,    689,   1564,   2993,    5156,    8257, ...
  1,  2117,   7553,  18679,  38705,   71801,  123217, ...
  1, 26830, 101961, 268714, 592489, 1166886, 2120545, ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• N. J. A. Sloane, Transforms

Crossrefs

Columns m=0-10 give: A000012, A088957, A089461, A089464, A218496, A218497, A218498, A218499, A218500, A218501, A218502.

Rows n=0-2 give: A000012, A000027, A028872.

Main diagonal gives A252766.

Cf. A007318, A088956.

Programs

Maple

hymtr:= proc(p) proc(n,m) `if`(m=0, p(n), m*add(
           p(k)*binomial(n, k) *(n-k+m)^(n-k-1), k=0..n))
        end end:
A:= hymtr(1):
seq(seq(A(n, d-n), n=0..d), d=0..12);

Mathematica

a[, 0] = 1; a[n, k_] := Sum[k*(n - j + k)^(n - j - 1)*Binomial[n, j], {j, 0, n}]; Table[a[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jun 24 2013 *)

Formula

E.g.f. of column k: exp(x) * (-LambertW(-x)/x)^k.

A(n,k) = Sum_{j=0..n} k * (n-j+k)^(n-j-1) * C(n,j).