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

0,5

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

Links

Alois P. Heinz, Rows n = 0..140, flattened

N. J. A. Sloane, Transforms

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).

Example

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, ...

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 *)

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.

Sequence in context: A307883 A111528 A363007 * A370072 A287024 A107702

Adjacent sequences: A144300 A144301 A144302 * A144304 A144305 A144306

Keyword

nonn,tabl

Author

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

Status

approved