A362369 Triangle read by rows, T(n, k) = binomial(n, k) * k! * Stirling2(n-k, k), for n >= 0 and 0 <= k <= n//2, where '//' denotes integer division.

1, 0, 0, 2, 0, 3, 0, 4, 12, 0, 5, 60, 0, 6, 210, 120, 0, 7, 630, 1260, 0, 8, 1736, 8400, 1680, 0, 9, 4536, 45360, 30240, 0, 10, 11430, 216720, 327600, 30240, 0, 11, 28050, 956340, 2772000, 831600, 0, 12, 67452, 3993000, 20207880, 13305600, 665280

Offset

0,4

Links

Table of n, a(n) for n=0..48.

Formula

From Mélika Tebni, May 10 2023: (Start)

E.g.f. of column k: (x*(exp(x)-1))^k / k!.

Sum_{k=0..n-1} (-1)^(n+k-1)*T(n+k-1, k) = A000169(n), for n > 0. (End)

Example

Triangle T(n, k) starts:
[0] 1;
[1] 0;
[2] 0, 2;
[3] 0, 3;
[4] 0, 4,   12;
[5] 0, 5,   60;
[6] 0, 6,  210,   120;
[7] 0, 7,  630,  1260;
[8] 0, 8, 1736,  8400,  1680;
[9] 0, 9, 4536, 45360, 30240;

Maple

T := (n, k) -> binomial(n, k) * k! * Stirling2(n-k, k):
seq(seq(T(n, k), k = 0..iquo(n, 2)), n = 0..9);
# second program:
egf := k-> (x*(exp(x)-1))^k / k!:
A362369 := (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A362369(n, k), k=0..iquo(n, 2))), n=0..12); # Mélika Tebni, May 10 2023

Prog

(SageMath)
def A362369(n, k):
    return binomial(n, k) * factorial(k) * stirling_number2(n - k, k)
for n in range(10):
    print([A362369(n, k) for k in range(n//2 + 1)])

Crossrefs

Cf. A000169, A052506 (row sums), A362788, A362789.

Sequence in context: A357475 A066682 A239968 * A320311 A240140 A240141

Adjacent sequences: A362366 A362367 A362368 * A362370 A362371 A362372

Keyword

nonn,tabf

Author

Peter Luschny, May 04 2023

Status

approved