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A357119
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} |Stirling1(n,k*j)|.
4
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, 12, 120, 0, 1, 0, 0, 0, 6, 60, 720, 0, 1, 0, 0, 0, 1, 35, 360, 5040, 0, 1, 0, 0, 0, 0, 10, 226, 2520, 40320, 0, 1, 0, 0, 0, 0, 1, 85, 1645, 20160, 362880, 0, 1, 0, 0, 0, 0, 0, 15, 735, 13454, 181440, 3628800, 0
OFFSET
0,9
LINKS
Eric Weisstein's World of Mathematics, Pochhammer Symbol.
FORMULA
For k > 0, e.g.f. of column k: Sum_{j>=0} (-log(1-x))^(k*j)/(k*j)!.
For k > 0, T(n,k) = ( Sum_{j=0..k-1} (w^j)_n )/k, where (x)_n is the Pochhammer symbol and w = exp(2*Pi*i/k).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 0, 0, 0, 0, 0, ...
0, 2, 1, 0, 0, 0, 0, ...
0, 6, 3, 1, 0, 0, 0, ...
0, 24, 12, 6, 1, 0, 0, ...
0, 120, 60, 35, 10, 1, 0, ...
0, 720, 360, 226, 85, 15, 1, ...
PROG
(PARI) T(n, k) = sum(j=0, n, abs(stirling(n, k*j, 1)));
(PARI) T(n, k) = if(k==0, 0^n, n!*polcoef(sum(j=0, n\k, (-log(1-x+x*O(x^n)))^(k*j)/(k*j)!), n));
(PARI) Pochhammer(x, n) = prod(k=0, n-1, x+k);
T(n, k) = if(k==0, 0^n, my(w=exp(2*Pi*I/k)); round(sum(j=0, k-1, Pochhammer(w^j, n)))/k);
CROSSREFS
Columns k=0-3 give: A000007, A000142, (-1)^n * A105752(n), A357828.
Cf. A357293.
Sequence in context: A276981 A230305 A357293 * A357883 A091728 A108069
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Oct 17 2022
STATUS
approved