|
%I #6 Feb 04 2019 11:22:55
%S 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 23,4,1,761,29531,801,1069,27,39,9,1,7129,6515,4523,285,3013,75,145,5,
%U 1,7381,177133,84095,341693,8591,7513,605,44,11,1,83711,190553,341747,139381,242537,1903,10831,33,35,6,1
%N 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 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 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 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 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 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 T(n,k) = numerator of H(n,k), where H(n,k) = ((1/n!)*(-1)^(r + 1))*(((d/dt)^n (1/t)*log(t)^(r + 1))_{t=1}).
%e The triangle H(n,k) begins:
%e n\k | 0 1 2 3 4 5 6
%e -----------------------------------------------------
%e 1 | 1
%e 2 | 3/2 1
%e 3 | 11/6 2 1
%e 4 | 25/12 35/12 5/2 1
%e 5 | 137/60 15/4 17/4 3 1
%e 6 | 49/20 203/45 49/8 35/6 7/2 1
%e 7 | 363/140 469/90 967/120 28/3 23/3 4 1
%e ...
%t H[n_, k_] := -(-1)^(n + k)/n!*(D[Log[t]^(k + 1)/t, {t, n}] /. t->1)
%t Table[Numerator[H[n, k]], {n, 1, 20}, {k, 0, n - 1}] // Flatten
%o (Maxima)
%o H(n, k) := -(-1)^(k + n)/n!*at(diff(log(t)^(k + 1)/t, t, n), t = 1)$
%o create_list(num(H(n, k)), n, 1, 20, k, 0, n - 1);
%Y Cf. A001008 (column 0), A323855 (denominators).
%K nonn,easy,tabl,frac
%O 1,2
%A _Franck Maminirina Ramaharo_, Feb 01 2019
|
