A367198 T(n, k) = Sum_{m = 0..n-1} Stirling1(m+1, k)*binomial(n, m)*(-1)^(n + k), where "Stirling1" are the signed Stirling numbers of the first kind.

1, 1, 2, 4, 6, 3, 15, 30, 18, 4, 76, 165, 125, 40, 5, 455, 1075, 930, 380, 75, 6, 3186, 8015, 7679, 3675, 945, 126, 7, 25487, 67536, 70042, 37688, 11550, 2044, 196, 8, 229384, 634935, 702372, 414078, 144417, 30870, 3990, 288, 9, 2293839, 6591943, 7696245, 4886390, 1885065, 463092, 73080, 7200, 405, 10

Offset: 1

Author

Thomas Scheuerle, Nov 10 2023

Keywords

• nonn
• tabl

Comments

To use the unsigned Stirling numbers rewrite the formula as: T(n, k) = Sum_{m = 0..n-1} abs(Stirling1(m+1, k))*binomial(n, m)*(-1)^(1+m+n). Replacing in this formula Stirling1 (A008275) by Stirling2 (A048993) one obtains a shifted version of A321331.

Examples

			Triangle begins:
    1;
    1,    2;
    4,    6,   3;
   15,   30,  18,   4;
   76,  165, 125,  40,  5;
  455, 1075, 930, 380, 75, 6;
		

Crossrefs

Cf. A002411, A002467 (first column), A000027 (main diagonal), A008275.

Cf. A180191(n+1) (row sums), A321331 (variant with Stirling2).

Programs

Maple

T := (n, k) -> local m; add(Stirling1(m+1, k)*binomial(n, m)*(-1)^(n + k), m = 0..n-1): seq(seq(T(n, k), k = 1..n), n = 1..9);  # Peter Luschny, Nov 10 2023

PARI

T(n,k) = sum(m=0, n-1, stirling(m+1, k)*binomial(n, m)*(-1)^(n+k))

Formula

T(n+1, n) = n^2*(n+1)/2 = A002411(n).

T(n, n-2) = 6*T(n-1, n-3) - 15*T(n-2, n-4) + 20*T(n-3, n-5) - 15*T(n-4, n-6) + 6*T(n-5, n-7) - T(n-6, n-8), for n > 8.

T(n, n-k) = (-1)^k*Sum_{m=0..n-1} Stirling1(m+1, n-k)*binomial(n, m).