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A322265 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = numerator of Sum_{j=1..n} 1/j^k.

%I A322265 #7 Feb 16 2025 08:33:57
%S A322265 1,1,2,1,3,3,1,5,11,4,1,9,49,25,5,1,17,251,205,137,6,1,33,1393,2035,
%T A322265 5269,49,7,1,65,8051,22369,256103,5369,363,8,1,129,47449,257875,
%U A322265 14001361,28567,266681,761,9,1,257,282251,3037465,806108207,14011361,9822481,1077749,7129,10
%N A322265 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = numerator of Sum_{j=1..n} 1/j^k.
%H A322265 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>
%F A322265 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 A322265 Square array begins:
%e A322265   1,       1,          1,              1,                  1,  ...
%e A322265   2,     3/2,        5/4,            9/8,              17/16,  ...
%e A322265   3,    11/6,      49/36,        251/216,          1393/1296,  ...
%e A322265   4,   25/12,    205/144,      2035/1728,        22369/20736,  ...
%e A322265   5,  137/60,  5269/3600,  256103/216000,  14001361/12960000,  ...
%t A322265 Table[Function[k, Numerator[Sum[1/j^k, {j, 1, n}]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten
%t A322265 Table[Function[k, Numerator[HarmonicNumber[n, k]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten
%t A322265 Table[Function[k, Numerator[SeriesCoefficient[PolyLog[k, x]/(1 - x), {x, 0, n}]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten
%Y A322265 Columns k=0..10 give A000027, A001008, A007406, A007408, A007410, A099828, A103345, A103347, A103349, A103351, A103716.
%Y A322265 Denominators are in A322266.
%Y A322265 Cf. A103438, A276485.
%K A322265 nonn,tabl,frac
%O A322265 1,3
%A A322265 _Ilya Gutkovskiy_, Dec 01 2018