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)|.

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

Author

Seiichi Manyama, Oct 17 2022

Keywords

• nonn
• tabl

Examples

			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, ...
		

Links

• Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Crossrefs

Columns k=0-3 give: A000007, A000142, (-1)^n * A105752(n), A357828.

Cf. A357293.

Programs

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);

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).