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).

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, 23, 4, 1, 761, 29531, 801, 1069, 27, 39, 9, 1, 7129, 6515, 4523, 285, 3013, 75, 145, 5, 1, 7381, 177133, 84095, 341693, 8591, 7513, 605, 44, 11, 1, 83711, 190553, 341747, 139381, 242537, 1903, 10831, 33, 35, 6, 1

Offset: 1

Author

Franck Maminirina Ramaharo, Feb 01 2019

Keywords

• nonn
• easy
• tabl
• frac

Comments

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).

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).

Examples

			The triangle H(n,k) begins:
  n\k |   0        1       2       3     4      5     6
  -----------------------------------------------------
    1 |   1
    2 |   3/2      1
    3 |  11/6      2       1
    4 |  25/12    35/12    5/2     1
    5 | 137/60    15/4    17/4     3     1
    6 |  49/20   203/45   49/8    35/6   7/2   1
    7 | 363/140  469/90  967/120  28/3  23/3   4     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.

Crossrefs

Cf. A001008 (column 0), A323855 (denominators).

Programs

Mathematica

H[n_, k_] := -(-1)^(n + k)/n!*(D[Log[t]^(k + 1)/t, {t, n}] /. t->1)
Table[Numerator[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(num(H(n, k)), n, 1, 20, k, 0, n - 1);

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

Formula

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}).