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A322266 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = denominator of Sum_{j=1..n} 1/j^k.

%I A322266 #6 Feb 16 2025 08:33:57
%S A322266 1,1,1,1,2,1,1,4,6,1,1,8,36,12,1,1,16,216,144,60,1,1,32,1296,1728,
%T A322266 3600,20,1,1,64,7776,20736,216000,3600,140,1,1,128,46656,248832,
%U A322266 12960000,24000,176400,280,1,1,256,279936,2985984,777600000,12960000,8232000,705600,2520,1
%N A322266 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = denominator of Sum_{j=1..n} 1/j^k.
%H A322266 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>
%F A322266 G.f. of column k: PolyLog(k,x)/(1 - x), where PolyLog() is the polylogarithm function (for rationals Sum_{j=1..n} 1/j^k).
%e A322266 Square array begins:
%e A322266   1,       1,          1,              1,                  1,  ...
%e A322266   2,     3/2,        5/4,            9/8,              17/16,  ...
%e A322266   3,    11/6,      49/36,        251/216,          1393/1296,  ...
%e A322266   4,   25/12,    205/144,      2035/1728,        22369/20736,  ...
%e A322266   5,  137/60,  5269/3600,  256103/216000,  14001361/12960000,  ...
%t A322266 Table[Function[k, Denominator[Sum[1/j^k, {j, 1, n}]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten
%t A322266 Table[Function[k, Denominator[HarmonicNumber[n, k]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten
%t A322266 Table[Function[k, Denominator[SeriesCoefficient[PolyLog[k, x]/(1 - x), {x, 0, n}]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten
%Y A322266 Columns k=0..10 give A000012, A002805, A007407, A007409, A007480, A069052, A103346, A103348, A103350, A103352, A103717.
%Y A322266 Numerators are in A322265.
%Y A322266 Cf. A103438, A276487.
%K A322266 nonn,tabl,frac
%O A322266 1,5
%A A322266 _Ilya Gutkovskiy_, Dec 01 2018