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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. 1

%I

%S 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 5269,49,7,1,65,8051,22369,256103,5369,363,8,1,129,47449,257875,

%U 14001361,28567,266681,761,9,1,257,282251,3037465,806108207,14011361,9822481,1077749,7129,10

%N 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>

%F 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 Square array begins:

%e 1, 1, 1, 1, 1, ...

%e 2, 3/2, 5/4, 9/8, 17/16, ...

%e 3, 11/6, 49/36, 251/216, 1393/1296, ...

%e 4, 25/12, 205/144, 2035/1728, 22369/20736, ...

%e 5, 137/60, 5269/3600, 256103/216000, 14001361/12960000, ...

%t Table[Function[k, Numerator[Sum[1/j^k, {j, 1, n}]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten

%t Table[Function[k, Numerator[HarmonicNumber[n, k]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten

%t Table[Function[k, Numerator[SeriesCoefficient[PolyLog[k, x]/(1 - x), {x, 0, n}]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten

%Y Columns k=0..10 give A000027, A001008, A007406, A007408, A007410, A099828, A103345, A103347, A103349, A103351, A103716.

%Y Denominators are in A322266.

%Y Cf. A103438, A276485.

%K nonn,tabl,frac

%O 1,3

%A _Ilya Gutkovskiy_, Dec 01 2018

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Last modified December 6 18:03 EST 2019. Contains 329809 sequences. (Running on oeis4.)