A294042 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp((1+x)^k - 1).

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 15, 20, 1, 0, 1, 5, 28, 87, 76, 1, 0, 1, 6, 45, 232, 585, 312, 1, 0, 1, 7, 66, 485, 2248, 4383, 1384, 1, 0, 1, 8, 91, 876, 6145, 24544, 35919, 6512, 1, 0, 1, 9, 120, 1435, 13716, 88245, 295456, 318195, 32400, 1, 0

Offset: 0

Author

Seiichi Manyama, Oct 22 2017

Keywords

• nonn
• tabl

Examples

			Square array A(n,k) begins:
   1, 1,   1,    1,     1,     1, ...
   0, 1,   2,    3,     4,     5, ...
   0, 1,   6,   15,    28,    45, ...
   0, 1,  20,   87,   232,   485, ...
   0, 1,  76,  585,  2248,  6145, ...
   0, 1, 312, 4383, 24544, 88245, ...
		

Links

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

Crossrefs

Columns k=0..5 give A000007, A000012, A000898, A192989, A202824, A202825.

Rows n=0..2 give A000012, A001477, A000384.

Main diagonal gives A294045.

Cf. A000110, A293991.

Formula

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

A(n,k) = Sum_{j=0..n} k^j * Stirling1(n,j) * Bell(j). - Seiichi Manyama, Jan 31 2024