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A355650
Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^k/k! * (exp(x) - 1)).
0
1, 1, 1, 1, 0, 2, 1, 0, 2, 5, 1, 0, 0, 3, 15, 1, 0, 0, 3, 16, 52, 1, 0, 0, 0, 6, 65, 203, 1, 0, 0, 0, 4, 10, 336, 877, 1, 0, 0, 0, 0, 10, 105, 1897, 4140, 1, 0, 0, 0, 0, 5, 20, 651, 11824, 21147, 1, 0, 0, 0, 0, 0, 15, 35, 2968, 80145, 115975, 1, 0, 0, 0, 0, 0, 6, 35, 616, 18936, 586000, 678570
OFFSET
0,6
FORMULA
T(0,k) = 1 and T(n,k) = ((n-1)!/k!) * Sum_{j=k+1..n} (j/(j-k)!) * T(n-j,k)/(n-j)! for n > 0.
T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} Stirling2(n-k*j,j)/(k!^j * (n-k*j)!).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 0, 0, 0, 0, 0, 0, ...
2, 2, 0, 0, 0, 0, 0, ...
5, 3, 3, 0, 0, 0, 0, ...
15, 16, 6, 4, 0, 0, 0, ...
52, 65, 10, 10, 5, 0, 0, ...
203, 336, 105, 20, 15, 6, 0, ...
PROG
(PARI) T(n, k) = n!*sum(j=0, n\(k+1), stirling(n-k*j, j, 2)/(k!^j*(n-k*j)!));
CROSSREFS
Columns k=0..3 give A000110, A052506, A354000, A354001.
Sequence in context: A146162 A147702 A118208 * A292892 A074142 A059084
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Jul 12 2022
STATUS
approved