|
|
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, 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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
LINKS
|
|
|
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, 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
|
|
CROSSREFS
|
Columns k=0..10 give A000027, A001008, A007406, A007408, A007410, A099828, A103345, A103347, A103349, A103351, A103716.
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|