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A357883
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (k*j)!* |Stirling1(n,k*j)|/(k!^j * j!).
0
1, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 6, 0, 1, 0, 0, 3, 24, 0, 1, 0, 0, 1, 14, 120, 0, 1, 0, 0, 0, 6, 80, 720, 0, 1, 0, 0, 0, 1, 35, 544, 5040, 0, 1, 0, 0, 0, 0, 10, 235, 4284, 40320, 0, 1, 0, 0, 0, 0, 1, 85, 1834, 38310, 362880, 0, 1, 0, 0, 0, 0, 0, 15, 735, 16352, 383256, 3628800, 0
OFFSET
0,9
FORMULA
For k > 0, e.g.f. of column k: exp((-log(1-x))^k / k!).
T(0,k) = 1; T(n,k) = Sum_{j=1..n} binomial(n-1,j-1) * |Stirling1(j,k)| * T(n-j,k).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 0, 0, 0, 0, ...
0, 2, 1, 0, 0, 0, ...
0, 6, 3, 1, 0, 0, ...
0, 24, 14, 6, 1, 0, ...
0, 120, 80, 35, 10, 1, ...
PROG
(PARI) T(n, k) = sum(j=0, n, (k*j)!*abs(stirling(n, k*j, 1))/(k!^j*j!));
(PARI) T(n, k) = if(k==0, 0^n, n!*polcoef(exp((-log(1-x+x*O(x^n)))^k/k!), n));
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Oct 18 2022
STATUS
approved