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, 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
Examples
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 1 1
...
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.
Programs
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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 -
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);
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PARI
T(n, k) = denominator(substvec(diffop(L^(k+1)/X, [L, X], [1/X, 1], n), [L, X], [0, 1])/n!); \\ Jinyuan Wang, Mar 13 2025
Formula
T(n,k) = denominator of H(n,k), where H(n,k) = ((1/n!)*(-1)^(n+k+1))*(((d/dt)^n (1/t)*log(t)^(k+1))_{t=1}).
Comments