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A323855
Triangle read by rows: T(n,k) is the denominator of the generalized harmonic number H(n,k) of rank k (n >= 1, 0 <= k <= n - 1).
1
1, 2, 1, 6, 1, 1, 12, 12, 2, 1, 60, 4, 4, 1, 1, 20, 45, 8, 6, 2, 1, 140, 90, 120, 3, 3, 1, 1, 280, 5040, 80, 80, 2, 4, 2, 1, 2520, 1008, 378, 16, 144, 4, 12, 1, 1, 2520, 25200, 6048, 15120, 288, 240, 24, 3, 2, 1, 27720, 25200, 21600, 5040, 6048, 40, 240, 1, 2, 1, 1
OFFSET
1,2
COMMENTS
See A323854 for the definition of H(n,k).
LINKS
Gi-Sang Cheon and Moawwad E. A. El-Mikkawy, Generalized harmonic number identities and a related matrix representation, J. Korean Math. Soc, Volume 44, 2007, 487-498.
Gi-Sang Cheon and Moawwad E. A. El-Mikkawy, Generalized harmonic numbers with Riordan arrays, Journal of Number Theory, Volume 128, Issue 2, 2008, 413-425.
Joseph M. Santmyer, A Stirling like sequence of rational numbers, Discrete Math., Volume 171, no. 1-3, 1997, 229-235, MR1454453.
FORMULA
T(n,k) = denominator of H(n,k), where H(n,k) = ((1/n!)*(-1)^(r + 1))*(((d/dt)^n (1/t)*log(t)^(r + 1))_{t=1}).
EXAMPLE
Triangle T(n,k) begins:
n\k | 0 1 2 3 4 5 6
---------------------------------------
1 | 1
2 | 2 1
3 | 6 1 1
4 | 12 12 2 1
5 | 60 4 4 3 1
6 | 20 45 8 6 2 1
7 | 140 90 120 3 3 4 1
...
MATHEMATICA
H[n_, k_] := -(-1)^(n + k)/n!*(D[Log[t]^(k + 1)/t, {t, n}] /. t->1)
Table[Denominator[H[n, k]], {n, 1, 20}, {k, 0, n - 1}] // Flatten
PROG
(Maxima)
H(n, k) := -(-1)^(k + n)/n!*at(diff(log(t)^(k + 1)/t, t, n), t = 1)$
create_list(denom(H(n, k)), n, 1, 20, k, 0, n - 1);
CROSSREFS
Cf. A323854 (numerators), A002805 (Column 0).
Sequence in context: A089808 A290318 A139547 * A126342 A345461 A229818
KEYWORD
nonn,easy,tabl,frac
AUTHOR
STATUS
approved