A131182 - OEIS

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

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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)