A356857 Triangle of numbers T(n,k) = (-1)^(n-k)*(n+1)!*Stirling2(n,k)/(k+1)

1, -3, 2, 12, -24, 6, -60, 280, -180, 24, 360, -3600, 4500, -1440, 120, -2520, 52080, -113400, 65520, -12600, 720, 20160, -846720, 3034080, -2822400, 940800, -120960, 5040, -181440, 15361920, -87635520, 123451776, -63504000, 13789440, -1270080, 40320

Offset

1,2

Comments

Coefficients of Sum_{k=1..n} T(n,k)*(m+k)!/((m-1)!*(n+1)!) = Sum_{i=1..m} i^n.

Links

Table of n, a(n) for n=1..36.

Formula

T(n,k) = (-1)^(n-k)*(n+1)!/(k+1)! * Sum_{i=0..k} (-1)^i*binomial(k,i)*(k-i)^n.

T(n+1,k+1) = (-1)^(n+k)*(n+2)*(k+1)*(abs(T(n,k)/(k+2)) + abs(T(n,k+1))) with T(n,1) = (-1)^(n+1)*(n+1)!/2 and T(n,k) = 0 if n < k.

Example

The triangle T(n,k) begins:
n\k       1         2          3          4          5         6         7      8
1:        1
2:       -3         2
3:       12       -24          6
4:      -60       280       -180         24
5:      360     -3600       4500      -1440        120
6:    -2520     52080    -113400      65520     -12600       720
7:    20160   -846720    3034080   -2822400     940800   -120960      5040
8:  -181440  15361920  -87635520  123451776  -63504000  13789440  -1270080  40320
---------------------------------------------------------------------------------

Mathematica

T[n_, k_] := (-1)^(n - k) * (n + 1)!/(k + 1) * StirlingS2[n, k]; Table[T[n, k], {n, 1, 8}, {k, 1, n}] // Flatten (* Amiram Eldar, Sep 01 2022 *)

Prog

(Python)
def A(d):
    A = [[0 for col in range(d)] for row in range(d)]
    for i in range(d):
        A[0][i] = 1
    for i in range(1, d):
        for j in range(i, d):
            A[i][j] = (A[i][j - 1] + A[i - 1][j - 1])*(i + 1)
    for i in range(d):
        for j in range(i, d):
            for k in range(i+3, j+3):
                A[i][j] *= k
    a = []
    for i in range(d):
        for j in range(i+1):
            a.append((-1)**(i+j)*A[j][i])
    return(a)
(PARI) T(n, k) = (-1)^(n-k)*(n+1)!*stirling(n, k, 2)/(k+1); \\ Michel Marcus, Sep 26 2022

Crossrefs

Closely related to A019538.

Sequence in context: A136657 A006774 A361865 * A086551 A254215 A007214

Adjacent sequences: A356854 A356855 A356856 * A356858 A356859 A356860

Keyword

sign,tabl

Author

Samuel Gantner, Aug 31 2022

Status

approved