cp's OEIS Frontend

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).

%I A323855 #10 Mar 19 2025 10:04:06
%S A323855 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,
%T A323855 280,5040,80,80,2,4,2,1,2520,1008,378,16,144,4,12,1,1,2520,25200,6048,
%U A323855 15120,288,240,24,3,2,1,27720,25200,21600,5040,6048,40,240,1,2,1,1
%N 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).
%C A323855 See A323854 for the definition of H(n,k).
%H A323855 Gi-Sang Cheon and Moawwad E. A. El-Mikkawy, <a href="https://doi.org/10.4134/JKMS.2007.44.2.487">Generalized harmonic number identities and a related matrix representation</a>, J. Korean Math. Soc, Volume 44, 2007, 487-498.
%H A323855 Gi-Sang Cheon and Moawwad E. A. El-Mikkawy, <a href="https://doi.org/10.1016/j.jnt.2007.08.011">Generalized harmonic numbers with Riordan arrays</a>, Journal of Number Theory, Volume 128, Issue 2, 2008, 413-425.
%H A323855 Joseph M. Santmyer, <a href="http://dx.doi.org/10.1016/S0012-365X(96)00082-9">A Stirling like sequence of rational numbers</a>, Discrete Math., Volume 171, no. 1-3, 1997, 229-235, MR1454453.
%F A323855 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}).
%e A323855 Triangle T(n,k) begins:
%e A323855   n\k |   0    1    2    3    4    5    6
%e A323855   ---------------------------------------
%e A323855     1 |   1
%e A323855     2 |   2    1
%e A323855     3 |   6    1    1
%e A323855     4 |  12   12    2    1
%e A323855     5 |  60    4    4    3    1
%e A323855     6 |  20   45    8    6    2    1
%e A323855     7 | 140   90  120    3    3    1    1
%e A323855     ...
%t A323855 H[n_, k_] := -(-1)^(n + k)/n!*(D[Log[t]^(k + 1)/t, {t, n}] /. t->1)
%t A323855 Table[Denominator[H[n, k]], {n, 1, 20}, {k, 0, n - 1}] // Flatten
%o A323855 (Maxima)
%o A323855 H(n, k) := -(-1)^(k + n)/n!*at(diff(log(t)^(k + 1)/t, t, n), t = 1)$
%o A323855 create_list(denom(H(n, k)), n, 1, 20, k, 0, n - 1);
%o A323855 (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
%Y A323855 Cf. A002805 (Column 0), A323854 (numerators).
%K A323855 nonn,easy,tabl,frac
%O A323855 1,2
%A A323855 _Franck Maminirina Ramaharo_, Feb 01 2019