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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.
0
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
1,3
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.
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).
EXAMPLE
Triangle begins:
1;
1, 2;
4, 6, 3;
15, 30, 18, 4;
76, 165, 125, 40, 5;
455, 1075, 930, 380, 75, 6;
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
PROG
(PARI) T(n, k) = sum(m=0, n-1, stirling(m+1, k)*binomial(n, m)*(-1)^(n+k))
CROSSREFS
Cf. A002411, A002467 (first column), A000027 (main diagonal), A008275.
Cf. A180191(n+1) (row sums), A321331 (variant with Stirling2).
Sequence in context: A348022 A072984 A339671 * A317310 A231655 A018841
KEYWORD
nonn,tabl
AUTHOR
Thomas Scheuerle, Nov 10 2023
STATUS
approved