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A323854
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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).
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2
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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
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OFFSET
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1,2
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COMMENTS
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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).
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LINKS
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FORMULA
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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}).
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EXAMPLE
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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
...
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MATHEMATICA
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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
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PROG
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(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);
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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