A323855 -id:A323855 - cp's OEIS frontend

A209155 Triangle read by rows: numerators of Stirling-like triangle of rational numbers.

-1, -1, 1, -1, 3, -1, -1, 1, -2, 1, -1, 5, -5, 5, -1, -1, 1, -5, 5, -3, 1, -1, 7, -7, 35, -7, 7, -1, -1, 4, -28, 7, -14, 14, -4, 1

Offset: 1

N. J. A. Sloane, Mar 05 2012

In Joseph M. Santmyer's paper, Table 1 (A209155/A209156) seems to be incorrect. The paper presents the formula c(n,r) = -(n/(r+1)) * (D(n-1,r-1)/D(n,r)) * c(n-1,r-1) for 1 < r < n-1, where D(n,r) = A323855(n,r). However, in Table 1, the value of c(5,2) is given as -5/3, which does not match the result obtained by applying the formula: -5/3 * D(4,1)/D(5,2) * c(4,1) equals -5. - Jinyuan Wang, Mar 17 2025

			Triangle begins:
  -1;
  -1,  1;
  -1, 3/2,    -1;
  -1,   1,    -2,    1;
  -1,   5,  -5/3,  5/2,   -1;
  -1,   1,    -5,  5/2,   -3,    1;
  -1,   7,  -7/3, 35/4, -7/2,  7/2, -1;
  -1,   4, -28/3,  7/3,  -14, 14/3, -4, 1;
  ...

Joseph M. Santmyer, A Stirling like sequence of rational numbers, Discrete Math. 171 (1997), no. 1-3, 229-235. MR1454453.

Cf. A209156, A323855.

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).

Original entry on oeis.org

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

Franck Maminirina Ramaharo, Feb 01 2019

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).

			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
    ...

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.

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

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

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);

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

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}).

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A209155 Triangle read by rows: numerators of Stirling-like triangle of rational numbers.

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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).

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