A292860 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(exp(x) - 1)).

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 5, 0, 1, 4, 12, 22, 15, 0, 1, 5, 20, 57, 94, 52, 0, 1, 6, 30, 116, 309, 454, 203, 0, 1, 7, 42, 205, 756, 1866, 2430, 877, 0, 1, 8, 56, 330, 1555, 5428, 12351, 14214, 4140, 0, 1, 9, 72, 497, 2850, 12880, 42356, 88563, 89918, 21147, 0

Offset

0,8

Links

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Formula

A(0,k) = 1 and A(n,k) = k * Sum_{j=0..n-1} binomial(n-1,j) * A(j,k) for n > 0.

A(n,k) = Sum_{j=0..n} k^j * Stirling2(n,j). - Seiichi Manyama, Jul 27 2019

A(n,k) = BellPolynomial(n, k). - Peter Luschny, Dec 23 2021

Example

Square array begins:
   1,   1,    1,     1,     1,      1,      1, ...
   0,   1,    2,     3,     4,      5,      6, ...
   0,   2,    6,    12,    20,     30,     42, ...
   0,   5,   22,    57,   116,    205,    330, ...
   0,  15,   94,   309,   756,   1555,   2850, ...
   0,  52,  454,  1866,  5428,  12880,  26682, ...
   0, 203, 2430, 12351, 42356, 115155, 268098, ...

Maple

A:= proc(n, k) option remember; `if`(n=0, 1,
      (1+add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 25 2017

Mathematica

A[0, _] = 1; A[n_ /; n >= 0, k_ /; k >= 0] := A[n, k] = k*Sum[Binomial[n-1, j]*A[j, k], {j, 0, n-1}]; A[_, _] = 0;
Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 13 2021 *)
A292860[n_, k_] := BellB[n, k]; Table[A292860[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Peter Luschny, Dec 23 2021 *)

Crossrefs

Columns k=0-10 give: A000007, A000110, A001861, A027710, A078944, A144180, A144223, A144263, A221159, A276506, A276507.

Rows n=0..2 give A000012, A001477, A002378.

Main diagonal gives A242817.

Same array, different indexing is A189233.

Cf. A292861.

Sequence in context: A351339 A361432 A294498 * A265609 A362125 A261718

Adjacent sequences: A292857 A292858 A292859 * A292861 A292862 A292863

Keyword

nonn,tabl

Author

Seiichi Manyama, Sep 25 2017

Status

approved