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
Examples
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).
Links
- 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.
Crossrefs
Programs
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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
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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
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)
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