cp's OEIS Frontend

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

%I A131182 #44 Sep 03 2022 10:42:28
%S A131182 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 A131182 720,3840,1944,384,50,6,1,0,5040,46080,29160,6144,750,72,7,1,0,40320,
%U A131182 645120,524880,122880,15000,1296,98,8,1,0,362880,10321920,11022480,2949120,375000,31104,2058,128,9,1
%N A131182 Table T(n,k) = n!*k^n, read by upwards antidiagonals.
%C A131182 For k>0, T(n,k) is the n-th moment of the exponential distribution with mean = k. - _Geoffrey Critzer_, Jan 06 2019
%C A131182 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 A131182 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 A131182 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 A131182 From _Ilya Gutkovskiy_, Aug 11 2017: (Start)
%F A131182 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 A131182 E.g.f. of column k: 1/(1 - k*x). (End)
%e A131182 The (inverted) table begins:
%e A131182 k=0: 1, 0,   0,    0,      0,       0, ... (A000007)
%e A131182 k=1: 1, 1,   2,    6,     24,     120, ... (A000142)
%e A131182 k=2: 1, 2,   8,   48,    384,    3840, ... (A000165)
%e A131182 k=3: 1, 3,  18,  162,   1944,   29160, ... (A032031)
%e A131182 k=4: 1, 4,  32,  384,   6144,  122880, ... (A047053)
%e A131182 k=5: 1, 5,  50,  750,  15000,  375000, ... (A052562)
%e A131182 k=6: 1, 6,  72, 1296,  31104,  933120, ... (A047058)
%e A131182 k=7: 1, 7,  98, 2058,  57624, 2016840, ... (A051188)
%e A131182 k=8: 1, 8, 128, 3072,  98304, 3932160, ... (A051189)
%e A131182 k=9: 1, 9, 162, 4374, 157464, 7085880, ... (A051232)
%e A131182 Main diagonal is 1, 1, 8, 162, 6144, 375000, ... (A061711).
%p A131182 T:= (n,k)-> n!*k^n:
%p A131182 seq(seq(T(d-k, k), k=0..d), d=0..12);  # _Alois P. Heinz_, Jan 06 2019
%o A131182 (Python)
%o A131182 from math import factorial
%o A131182 def A131182_T(n, k): # compute T(n, k)
%o A131182     return factorial(n)*k**n # _Chai Wah Wu_, Sep 01 2022
%Y A131182 Columns k=0-9 give: A000007, A000142, A000165, A032031, A047053, A052562, A047058, A051188, A051189, A051232.
%Y A131182 Main diagonal gives A061711.
%Y A131182 Cf. A292783, A303489, A331988.
%K A131182 nonn,tabl
%O A131182 0,8
%A A131182 _Philippe Deléham_, Sep 25 2007