Takao Komatsu - cp's OEIS frontend

A219247 Denominators of poly-Cauchy numbers of the second kind hat c_n^(2).

1, 4, 36, 48, 1800, 240, 35280, 20160, 226800, 50400, 3659040, 665280, 1967565600, 2242240, 129729600, 34594560, 2677989600, 66830400, 1857684628800, 39109150080, 3226504881600, 307286179200, 2333316585600, 1285014931200, 2192556726360000, 25057791158400

Offset: 0

Takao Komatsu, Mar 31 2013

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

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. A002790, A223899, A224102 (numerators).

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

a(n) = denominator(sum(k=0, n, stirling(n, k, 1)*(-1)^k/(k+1)^2)); \\ Michel Marcus, Nov 14 2015

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

Original entry on oeis.org

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

A224107 Denominators of poly-Cauchy numbers of the second kind hat c_n^(5).

Original entry on oeis.org

1, 32, 7776, 82944, 388800000, 155520000, 2613824640000, 11948912640000, 3629482214400000, 806551603200000, 77937565348177920000, 14170466426941440000, 92074412343521441433600000, 524640526173911347200000, 6070840374298117017600000, 12951126131835982970880000

Offset: 0

Takao Komatsu, Mar 31 2013

The poly-Cauchy numbers of the second kind hat c_n^(k) can be expressed in terms of the (unsigned) Stirling numbers of the first kind: hat c_n^(k) = (-1)^n*sum(abs(stirling1(n,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.
Takao 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. A002790, A223904, A219247, A224103, A224105, A224109 (numerators).

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

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

More terms from 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

A224096 Denominators of poly-Cauchy numbers c_n^(3).

Original entry on oeis.org

1, 8, 216, 576, 108000, 14400, 14817600, 16934400, 571536000, 127008000, 101428588800, 18441561600, 709031939616000, 12120204096000, 6678479808000, 24932991283200, 229679599076928000, 818822100096000

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. A006233, A222636, A224094, A224097 (numerators).

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

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

A224105 Denominators of poly-Cauchy numbers of the second kind hat c_n^(4).

Original entry on oeis.org

1, 16, 1296, 6912, 6480000, 288000, 6223392000, 14224896000, 1440270720000, 64012032000, 562320096307200, 511200087552000, 255506749760021760000, 1455878916011520000, 673863955411046400, 17969705477627904000

Offset: 0

Takao Komatsu, Mar 31 2013

The poly-Cauchy numbers of the second kind hat c_n^(k) can be expressed in terms of the (unsigned) Stirling numbers of the first kind: hat c_n^(k) = (-1)^n*sum(abs(stirling1(n,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. A002790, A223902, A219247, A224103, A224106 (numerators).

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

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

A224098 Denominators of poly-Cauchy numbers c_n^(4).

Original entry on oeis.org

1, 16, 1296, 6912, 6480000, 2592000, 6223392000, 14224896000, 1440270720000, 320060160000, 2811600481536000, 511200087552000, 255506749760021760000, 291175783202304000, 16846598885276160000

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, A222748, A224094, A224096, A224099 (numerators).

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

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

A224106 Numerators of poly-Cauchy numbers of the second kind hat c_n^(4).

Original entry on oeis.org

1, -1, 97, -1147, 3472243, -653983, 74118189437, -1058923294571, 777910456216513, -285577840060819, 23240203016832136201, -216925341603548096639, 1222007019804929270080450811

Offset: 0

Takao Komatsu, Mar 31 2013

The poly-Cauchy numbers of the second kind hat c_n^(k) can be expressed in terms of the (unsigned) Stirling numbers of the first kind: hat c_n^(k) = (-1)^n*sum(abs(stirling1(n,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. A002657, A223902, A224105, A114102, A224104, A224105 (denominators).

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

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

A224104 Numerators of poly-Cauchy numbers of the second kind hat c_n^(3).

Original entry on oeis.org

1, -1, 35, -217, 135989, -236881, 435876493, -3174551347, 790667708347, -1473406853309, 11050163107919893, -20886680047664287, 9154917271574968829623, -277315386220087376401, 803143323197313772705

Offset: 0

Takao Komatsu, Mar 31 2013

The poly-Cauchy numbers of the second kind hat c_n^(k) can be expressed in terms of the (unsigned) Stirling numbers of the first kind: hat c_n^(k) = (-1)^n*sum(abs(stirling1(n,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. A002657, A223901, A224102, A224103 (denominators).

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

a(n) = numerator(sum(k=0, n, stirling(n, k, 1)*(-1)^k/(k+1)^3)); \\ Michel Marcus, Nov 14 2015

A224109 Numerators of poly-Cauchy numbers of the second kind hat c_n^(5).

Original entry on oeis.org

1, -1, 275, -6289, 92902541, -154473289, 13399738273333, -377635608584803, 822223497000264427, -1492945924219675973, 1323386773861946436609781, -2448418399924413951578983, 177825546947844845937070681472647

Offset: 0

Takao Komatsu, Mar 31 2013

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

Vincenzo Librandi, Table of n, a(n) for n = 0..260
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.
Takao 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. A002657, A223904, A224107, A224102, A224104, A224106, A224107 (denominators).

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

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

cp's OEIS Frontend

User: Takao Komatsu

A219247 Denominators of poly-Cauchy numbers of the second kind hat c_n^(2).

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

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A224107 Denominators of poly-Cauchy numbers of the second kind hat c_n^(5).

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

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A224096 Denominators of poly-Cauchy numbers c_n^(3).

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A224105 Denominators of poly-Cauchy numbers of the second kind hat c_n^(4).

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A224098 Denominators of poly-Cauchy numbers c_n^(4).

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A224106 Numerators of poly-Cauchy numbers of the second kind hat c_n^(4).

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