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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).
6
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
0,8
LINKS
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
EXAMPLE
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, ...
CROSSREFS
Columns k=0..5 give A000007, A000012, A000898, A192989, A202824, A202825.
Rows n=0..2 give A000012, A001477, A000384.
Main diagonal gives A294045.
Sequence in context: A118350 A361950 A183135 * A287316 A322280 A331436
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Oct 22 2017
STATUS
approved