A131182 - OEIS

%I #44 Sep 03 2022 10:42:28

%S 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,

%T 720,3840,1944,384,50,6,1,0,5040,46080,29160,6144,750,72,7,1,0,40320,

%U 645120,524880,122880,15000,1296,98,8,1,0,362880,10321920,11022480,2949120,375000,31104,2058,128,9,1

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

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

%C 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

%H Chai Wah Wu, <a href="http://arxiv.org/abs/1508.02934">Permutations r_j such that ∑i∏j r_j(i) is maximized or minimized</a>, arXiv:1508.02934 [math.CO], 2015-2020.

%H Chai Wah Wu, <a href="https://arxiv.org/abs/2002.10514">On rearrangement inequalities for multiple sequences</a>, arXiv:2002.10514 [math.CO], 2020.

%F From _Ilya Gutkovskiy_, Aug 11 2017: (Start)

%F 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.

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

%e The (inverted) table begins:

%e k=0: 1, 0,   0,    0,      0,       0, ... (A000007)

%e k=1: 1, 1,   2,    6,     24,     120, ... (A000142)

%e k=2: 1, 2,   8,   48,    384,    3840, ... (A000165)

%e k=3: 1, 3,  18,  162,   1944,   29160, ... (A032031)

%e k=4: 1, 4,  32,  384,   6144,  122880, ... (A047053)

%e k=5: 1, 5,  50,  750,  15000,  375000, ... (A052562)

%e k=6: 1, 6,  72, 1296,  31104,  933120, ... (A047058)

%e k=7: 1, 7,  98, 2058,  57624, 2016840, ... (A051188)

%e k=8: 1, 8, 128, 3072,  98304, 3932160, ... (A051189)

%e k=9: 1, 9, 162, 4374, 157464, 7085880, ... (A051232)

%e Main diagonal is 1, 1, 8, 162, 6144, 375000, ... (A061711).

%p T:= (n,k)-> n!*k^n:

%p seq(seq(T(d-k, k), k=0..d), d=0..12);  # _Alois P. Heinz_, Jan 06 2019

%o (Python)

%o from math import factorial

%o def A131182_T(n, k): # compute T(n, k)

%o     return factorial(n)*k**n # _Chai Wah Wu_, Sep 01 2022

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

%Y Main diagonal gives A061711.

%Y Cf. A292783, A303489, A331988.

%K nonn,tabl

%O 0,8

%A _Philippe Deléham_, Sep 25 2007