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, 1, 2, 1, 3, 3, 1, 5, 11, 4, 1, 9, 49, 25, 5, 1, 17, 251, 205, 137, 6, 1, 33, 1393, 2035, 5269, 49, 7, 1, 65, 8051, 22369, 256103, 5369, 363, 8, 1, 129, 47449, 257875, 14001361, 28567, 266681, 761, 9, 1, 257, 282251, 3037465, 806108207, 14011361, 9822481, 1077749, 7129, 10
Offset: 1
Examples
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, ...
Links
- Eric Weisstein's World of Mathematics, Harmonic Number
Crossrefs
Programs
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Mathematica
Table[Function[k, Numerator[Sum[1/j^k, {j, 1, n}]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten Table[Function[k, Numerator[HarmonicNumber[n, k]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten Table[Function[k, Numerator[SeriesCoefficient[PolyLog[k, x]/(1 - x), {x, 0, n}]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten
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).