A131182 Table T(n,k) = n!*k^n, read by upwards antidiagonals.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 6, 8, 3, 1, 0, 24, 48, 18, 4, 1, 0, 120, 384, 162, 32, 5, 1, 0, 720, 3840, 1944, 384, 50, 6, 1, 0, 5040, 46080, 29160, 6144, 750, 72, 7, 1, 0, 40320, 645120, 524880, 122880, 15000, 1296, 98, 8, 1, 0, 362880, 10321920, 11022480, 2949120, 375000, 31104, 2058, 128, 9, 1

Offset

0,8

Comments

For k>0, T(n,k) is the n-th moment of the exponential distribution with mean = k. - Geoffrey Critzer, Jan 06 2019

T(n,k) is the minimum value of Product_{i=1..n} Sum_{j=1..k} r_j[i] where each r_j is a permutation of {1..n}. For the maximum value, see A331988. - Chai Wah Wu, Sep 01 2022

Links

Table of n, a(n) for n=0..65.

Chai Wah Wu, Permutations r_j such that ∑i∏j r_j(i) is maximized or minimized, arXiv:1508.02934 [math.CO], 2015-2020.

Chai Wah Wu, On rearrangement inequalities for multiple sequences, arXiv:2002.10514 [math.CO], 2020.

Formula

From Ilya Gutkovskiy, Aug 11 2017: (Start)

G.f. of column k: 1/(1 - k*x/(1 - k*x/(1 - 2*k*x/(1 - 2*k*x/(1 - 3*k*x/(1 - 3*k*x/(1 - ...))))))), a continued fraction.

E.g.f. of column k: 1/(1 - k*x). (End)

Example

The (inverted) table begins:
k=0: 1, 0,   0,    0,      0,       0, ... (A000007)
k=1: 1, 1,   2,    6,     24,     120, ... (A000142)
k=2: 1, 2,   8,   48,    384,    3840, ... (A000165)
k=3: 1, 3,  18,  162,   1944,   29160, ... (A032031)
k=4: 1, 4,  32,  384,   6144,  122880, ... (A047053)
k=5: 1, 5,  50,  750,  15000,  375000, ... (A052562)
k=6: 1, 6,  72, 1296,  31104,  933120, ... (A047058)
k=7: 1, 7,  98, 2058,  57624, 2016840, ... (A051188)
k=8: 1, 8, 128, 3072,  98304, 3932160, ... (A051189)
k=9: 1, 9, 162, 4374, 157464, 7085880, ... (A051232)
Main diagonal is 1, 1, 8, 162, 6144, 375000, ... (A061711).

Maple

T:= (n, k)-> n!*k^n:
seq(seq(T(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Jan 06 2019

Prog

(Python)
from math import factorial
def A131182_T(n, k): # compute T(n, k)
    return factorial(n)*k**n # Chai Wah Wu, Sep 01 2022

Crossrefs

Columns k=0-9 give: A000007, A000142, A000165, A032031, A047053, A052562, A047058, A051188, A051189, A051232.

Main diagonal gives A061711.

Cf. A292783, A303489, A331988.

Sequence in context: A130167 A084938 A135898 * A254883 A266599 A327365

Adjacent sequences: A131179 A131180 A131181 * A131183 A131184 A131185

Keyword

nonn,tabl

Author

Philippe Deléham, Sep 25 2007

Status

approved