A224095 -id:A224095 - cp's OEIS frontend

A224094 Denominators of poly-Cauchy numbers c_n^(2).

1, 4, 36, 48, 1800, 720, 35280, 20160, 226800, 10080, 731808, 665280, 1967565600, 11211200, 129729600, 34594560, 18745927200, 28641600, 371536925760, 3990729600, 3226504881600, 4877558400, 466663317120, 550720684800, 2192556726360000, 175404538108800

Offset: 0

Takao Komatsu, Mar 30 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. A006233, A222627, A224095 (numerators).

Table[Denominator[Sum[StirlingS1[n, k]/ (k + 1)^2, {k, 0, n}]], {n, 0, 25}]

a(n) = denominator(sum(k=0, n,stirling(n, k, 1)/(k+1)^2)); \\ Michel Marcus, Nov 15 2015

A224097 Numerators of poly-Cauchy numbers c_n^(3).

Original entry on oeis.org

1, 1, -19, 89, -46261, 23323, -114895757, 760567603, -174446569403, 302339104957, -2125170096355349, 3788248001789087, -1573899862241140688567, 317684785943639774839, -2242333884754953400123

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, A222636, A224095, A224096 (denominators).

Table[Numerator[Sum[StirlingS1[n, k]/ (k + 1)^3, {k, 0, n}]], {n, 0, 25}]

a(n) = numerator(sum(k=0, n,stirling(n, k, 1)/(k+1)^3)); \\ Michel Marcus, Nov 15 2015

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

Original entry on oeis.org

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

A224101 Numerators of poly-Cauchy numbers c_n^(5).

Original entry on oeis.org

1, 1, -211, 4241, -57453709, 29825987, -7362684132917, 198504470798947, -415989828245529323, 730328251215062341, -628191544925589374756597, 1131010588175721446183783, -80125844020238574218022657310343

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..250
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. A006232, A223023, A224095, A224097, A224099, A224100 (denominators).

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

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

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A224094 Denominators of poly-Cauchy numbers c_n^(2).

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A224097 Numerators of poly-Cauchy numbers c_n^(3).

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

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A224101 Numerators of poly-Cauchy numbers c_n^(5).

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