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.

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, 3600, 20, 1, 1, 64, 7776, 20736, 216000, 3600, 140, 1, 1, 128, 46656, 248832, 12960000, 24000, 176400, 280, 1, 1, 256, 279936, 2985984, 777600000, 12960000, 8232000, 705600, 2520, 1

Offset

1,5

Links

Table of n, a(n) for n=1..55.

Eric Weisstein's World of Mathematics, Harmonic Number

Formula

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).

Example

Square array begins:
  1,       1,          1,              1,                  1,  ...
  2,     3/2,        5/4,            9/8,              17/16,  ...
  3,    11/6,      49/36,        251/216,          1393/1296,  ...
  4,   25/12,    205/144,      2035/1728,        22369/20736,  ...
  5,  137/60,  5269/3600,  256103/216000,  14001361/12960000,  ...

Mathematica

Table[Function[k, Denominator[Sum[1/j^k, {j, 1, n}]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten
Table[Function[k, Denominator[HarmonicNumber[n, k]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten
Table[Function[k, Denominator[SeriesCoefficient[PolyLog[k, x]/(1 - x), {x, 0, n}]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten

Crossrefs

Columns k=0..10 give A000012, A002805, A007407, A007409, A007480, A069052, A103346, A103348, A103350, A103352, A103717.

Numerators are in A322265.

Cf. A103438, A276487.

Sequence in context: A255009 A156579 A351790 * A190284 A327639 A273891

Adjacent sequences: A322263 A322264 A322265 * A322267 A322268 A322269

Keyword

nonn,tabl,frac

Author

Ilya Gutkovskiy, Dec 01 2018

Status

approved