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.

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

Links

Table of n, a(n) for n=0..35.

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

Adjacent sequences: A357364 A357365 A357366 * A357368 A357369 A357370

Keyword

nonn,tabl

Author

Peter Luschny, Sep 26 2022

Status

approved