A224098 -id:A224098 - cp's OEIS frontend

A224099 Numerators of poly-Cauchy numbers c_n^(4).

1, 1, -65, 635, -1691507, 2602903, -30316306813, 405644259179, -281598937164737, 491752927006687, -38273845811539969069, 68624716189056755839, -372590717516807448774422779

Offset: 0

Takao Komatsu, Mar 31 2013

The poly-Cauchy numbers c_n^(k) can be expressed in terms of the (unsigned) Stirling numbers of the first kind: c_n^(k) = (-1)^n*sum(abs(stirling1(n,m))*(-1)^m/(m+1)^k, m=0..n).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Takao Komatsu, Poly-Cauchy numbers, RIMS Kokyuroku 1806 (2012)
Takao Komatsu, Poly-Cauchy numbers with a q parameter, Ramanujan J. 31 (2013), 353-371.
Takao Komatsu, Poly-Cauchy numbers, Kyushu J. Math. 67 (2013), 143-153.
T. Komatsu, V. Laohakosol, K. Liptai, A generalization of poly-Cauchy numbers and its properties, Abstract and Applied Analysis, Volume 2013, Article ID 179841, 8 pages.
Takao Komatsu, FZ Zhao, The log-convexity of the poly-Cauchy numbers, arXiv preprint arXiv:1603.06725, 2016

Cf. A006232, A222748, A224095, A224097, A224098 (denominators).

Table[Numerator[Sum[StirlingS1[n, k]/ (k + 1)^4, {k, 0, n}]], {n, 0, 25}]

a(n) = numerator(sum(k=0, n,stirling(n, k, 1)/(k+1)^4)); \\ Michel Marcus, Nov 15 2015

A224100 Denominators of poly-Cauchy numbers c_n^(5).

Original entry on oeis.org

1, 32, 7776, 82944, 388800000, 51840000, 2613824640000, 11948912640000, 3629482214400000, 806551603200000, 77937565348177920000, 14170466426941440000, 92074412343521441433600000

Offset: 0

Takao Komatsu, Mar 31 2013

The poly-Cauchy numbers c_n^(k) can be expressed in terms of the (unsigned) Stirling numbers of the first kind: c_n^(k) = (-1)^n*sum(abs(stirling1(n,m))*(-1)^m/(m+1)^k, m=0..n).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Takao Komatsu, Poly-Cauchy numbers, RIMS Kokyuroku 1806 (2012)
Takao Komatsu, Poly-Cauchy numbers with a q parameter, Ramanujan J. 31 (2013), 353-371.
Takao Komatsu, Poly-Cauchy numbers, Kyushu J. Math. 67 (2013), 143-153.
T. Komatsu, V. Laohakosol, and K. Liptai, A generalization of poly-Cauchy numbers and its properties, Abstract and Applied Analysis, Volume 2013, Article ID 179841, 8 pages.
Takao Komatsu and F.-Z. Zhao, The log-convexity of the poly-Cauchy numbers, arXiv preprint arXiv:1603.06725 [math.NT], 2016.

Cf. A006233, A223023, A224094, A224096, A224098, A224101 (numerators).

Table[Denominator[Sum[StirlingS1[n, k]/ (k + 1)^5, {k, 0, n}]], {n, 0, 25}]

a(n) = denominator(sum(k=0, n, stirling(n, k, 1)/(k+1)^5)); \\ Michel Marcus, Nov 15 2015

cp's OEIS Frontend

A224099 Numerators of poly-Cauchy numbers c_n^(4).

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