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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.
1
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
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
Numerators are in A322265.
Sequence in context: A255009 A156579 A351790 * A190284 A327639 A273891
KEYWORD
nonn,tabl,frac
AUTHOR
Ilya Gutkovskiy, Dec 01 2018
STATUS
approved