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A357367
Triangle read by rows. T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * L(n + m, m), where L denotes the unsigned Lah numbers A271703.
0
1, 0, 2, 0, 6, 12, 0, 24, 120, 120, 0, 120, 1080, 2520, 1680, 0, 720, 10080, 40320, 60480, 30240, 0, 5040, 100800, 604800, 1512000, 1663200, 665280, 0, 40320, 1088640, 9072000, 33264000, 59875200, 51891840, 17297280
OFFSET
0,3
FORMULA
T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * binomial(n + m - 1, m - 1) * (n + m)! / m!.
EXAMPLE
Triangle T(n, k) start:
[0] 1;
[1] 0, 2;
[2] 0, 6, 12;
[3] 0, 24, 120, 120;
[4] 0, 120, 1080, 2520, 1680;
[5] 0, 720, 10080, 40320, 60480, 30240;
[6] 0, 5040, 100800, 604800, 1512000, 1663200, 665280;
[7] 0, 40320, 1088640, 9072000, 33264000, 59875200, 51891840, 17297280;
MAPLE
T := (n, k) -> add((-1)^(m + k) * binomial(n + k, n + m) * binomial(n + m - 1, m - 1) * (n + m)! / m!, m = 0..k):
seq(print(seq(T(n, k), k = 0..n)), n = 0..8);
PROG
(SageMath)
def Lah(n, k): return binomial(n, k) * falling_factorial(n - 1, n - k)
def T(n, k): return (sum((-1)^(m + k) * binomial(n + k, n + m) * Lah(n + m, m)
for m in range(k + 1)))
for n in range(8): print([T(n, k) for k in range(n+1)])
CROSSREFS
Cf. A032037 (row sums), A271703.
Sequence in context: A362186 A335061 A350462 * A110667 A347929 A129877
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Sep 26 2022
STATUS
approved