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
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
Programs
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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).
Comments