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A323854 Triangle read by rows: T(n,k) is the numerator of the generalized harmonic number H(n,k) of rank k (n >= 1, 0 <= k <= n - 1).

%I A323854 #9 Mar 14 2025 21:17:42
%S A323854 1,3,1,11,2,1,25,35,5,1,137,15,17,3,1,49,203,49,35,7,1,363,469,967,28,
%T A323854 23,4,1,761,29531,801,1069,27,39,9,1,7129,6515,4523,285,3013,75,145,5,
%U A323854 1,7381,177133,84095,341693,8591,7513,605,44,11,1,83711,190553,341747,139381,242537,1903,10831,33,35,6,1
%N A323854 Triangle read by rows: T(n,k) is the numerator of the generalized harmonic number H(n,k) of rank k (n >= 1, 0 <= k <= n - 1).
%C A323854 Santmyer (1997) defined the generalized harmonic numbers H(n,k) of rank k by H(n,k) = Sum_{n_0 + n_1 + ... + n_k <= n} 1/(n_0*n_1*...*n_k).
%C A323854 If n >= 0, then the triangle {A323854(n+1,k)/A323855(n+1,k)}_{n,k} is the Riordan array (-log(1 - x)/(x*(1 - x)), -log(1 - x)/x).
%H A323854 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 A323854 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 A323854 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 A323854 T(n,k) = numerator 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 A323854 The triangle H(n,k) begins:
%e A323854   n\k |   0        1       2       3     4      5     6
%e A323854   -----------------------------------------------------
%e A323854     1 |   1
%e A323854     2 |   3/2      1
%e A323854     3 |  11/6      2       1
%e A323854     4 |  25/12    35/12    5/2     1
%e A323854     5 | 137/60    15/4    17/4     3     1
%e A323854     6 |  49/20   203/45   49/8    35/6   7/2   1
%e A323854     7 | 363/140  469/90  967/120  28/3  23/3   4     1
%e A323854     ...
%t A323854 H[n_, k_] := -(-1)^(n + k)/n!*(D[Log[t]^(k + 1)/t, {t, n}] /. t->1)
%t A323854 Table[Numerator[H[n, k]], {n, 1, 20}, {k, 0, n - 1}] // Flatten
%o A323854 (Maxima)
%o A323854 H(n, k) := -(-1)^(k + n)/n!*at(diff(log(t)^(k + 1)/t, t, n), t = 1)$
%o A323854 create_list(num(H(n, k)), n, 1, 20, k, 0, n - 1);
%o A323854 (PARI) T(n, k) = -(-1)^(n+k)*numerator(substvec(diffop(L^(k+1)/X, [L, X], [1/X, 1], n), [L, X], [0, 1])/n!); \\ _Jinyuan Wang_, Mar 13 2025
%Y A323854 Cf. A001008 (column 0), A323855 (denominators).
%K A323854 nonn,easy,tabl,frac
%O A323854 1,2
%A A323854 _Franck Maminirina Ramaharo_, Feb 01 2019