A222627 -id:A222627 - cp's OEIS frontend

A223899 Poly-Cauchy numbers of the second kind hat c_n^(-2).

1, -4, 13, -51, 244, -1392, 9260, -70508, 605320, -5788008, 61021872, -703384272, 8801449344, -118828732032, 1721888828928, -26656798602240, 439110126743040, -7669109089082880, 141557837068938240, -2753560001544053760, 56299265625742848000

Offset: 0

Takao Komatsu, Mar 29 2013

sign

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Takao Komatsu, Poly-Cauchy numbers with a q parameter, Ramanujan J. 31 (2013), 353-371.
Takao Komatsu, Poly-Cauchy numbers, RIMS Kokyuroku 1806 (2012), p. 42-53.
Takao Komatsu, Poly-Cauchy numbers, Kyushu J. Math. 67 (2013), 143-153.

Cf. A081046, A223901, A223902, A223904.

Cf. A222627.

[&+[StirlingFirst(n, k)*(-1)^k*(k+1)^2: k in [0..n]]: n in [0..23]]; // Vincenzo Librandi, Aug 03 2013

Table[Sum[StirlingS1[n, k] (-1)^k (k + 1)^2, {k, 0, n}], {n, 0, 30}]

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

a(n) = Sum_{k=0..n} (-1)^k * (k+1)^2 * Stirling1(n,k).
E.g.f.: (1 - log(1 + x) * (3 - log(1 + x))) / (1 + x). - Ilya Gutkovskiy, Aug 09 2021
From Seiichi Manyama, Apr 15 2025: (Start)
E.g.f.: Sum_{k>=0} (k+1)^2 * (-log(1+x))^k / k!.
a(n) = (-1)^n * Sum_{k=0..2} k! * Stirling2(3,k+1) * |Stirling1(n+1,k+1)|. (End)

A222636 Poly-Cauchy numbers c_n^(-3).

Original entry on oeis.org

1, 8, 19, -1, -10, 48, -234, 1302, -8328, 60672, -497688, 4547448, -45846864, 505862064, -6065584128, 78555965184, -1093053332736, 16264215348480, -257730606190080, 4333624828853760, -77067187081620480, 1445257352902763520, -28505367984508416000

Offset: 0

Takao Komatsu, Mar 28 2013

sign

Definition of poly-Cauchy numbers in A222627.

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, Some recurrence relations of poly-Cauchy numbers, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829-845.
M. Z. Spivey,Combinatorial sums and finite differences, Discr. Math. 307 (24) (2007) 3130-3146.
Wikipedia, Stirling transform

Column k=3 of A383049.

Cf. A223901.

[&+[StirlingFirst(n,k)*(k+1)^3: k in [0..n]]: n in [0..25]]; // Bruno Berselli, Mar 28 2013

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

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

a(n) = Sum_{k=0..n} Stirling1(n,k) * (k+1)^3.
E.g.f.: (1 + x) * (1 + 7 * log(1 + x) + 6 * log(1 + x)^2 + log(1 + x)^3). - Ilya Gutkovskiy, Aug 10 2021
E.g.f.: Sum_{k>=0} (k+1)^3 * log(1+x)^k / k!. - Seiichi Manyama, Apr 14 2025

A222748 Poly-Cauchy numbers c_n^(-4).

Original entry on oeis.org

1, 16, 65, 45, -116, 340, -1240, 5480, -28464, 169248, -1125840, 8197680, -63806016, 514314240, -4058967744, 26952984000, -37203513984, -4251686488704, 140692872720384, -3560137793538048, 84004474130786304, -1955196907518928896, 45927815909901004800

Offset: 0

Takao Komatsu, Mar 28 2013

sign

Definition of poly-Cauchy numbers in A222627.

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, Some recurrence relations of poly-Cauchy numbers, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829-845.
M. Z. Spivey,Combinatorial sums and finite differences, Discr. Math. 307 (24) (2007) 3130-3146
Wikipedia, Stirling transform

Column k=4 of A383049.

[&+[StirlingFirst(n,k)*(k+1)^4: k in [0..n]]: n in [0..25]]; // Bruno Berselli, Mar 28 2013

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

a(n) = sum(k=0, n, stirling(n, k, 1)*(k+1)^4); \\ Michel Marcus, Nov 14 2015

a(n) = Sum_{k=0..n} Stirling1(n,k) * (k+1)^4.
From Seiichi Manyama, Apr 14 2025: (Start)
E.g.f.: Sum_{k>=0} (k+1)^4 * log(1+x)^k / k!.
E.g.f.: (1+x) * Sum_{k=0..4} Stirling2(5,k+1) * log(1+x)^k. (End)

A223023 Poly-Cauchy numbers c_n^(-5).

Original entry on oeis.org

1, 32, 211, 359, -538, 984, -1866, 1110, 32640, -449760, 5035200, -55896960, 646005600, -7896549120, 102604234080, -1418189492640, 20828546505600, -324419255412480, 5346952977432960, -93035974518691200, 1705088403923592960, -32842738382065931520

Offset: 0

Takao Komatsu, Mar 28 2013

sign

Definition of poly-Cauchy numbers in A222627.

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, Some recurrence relations of poly-Cauchy numbers, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829-845.
M. Z. Spivey,Combinatorial sums and finite differences, Discr. Math. 307 (24) (2007) 3130-3146
Wikipedia, Stirling transform

Column k=5 of A383049.

[&+[StirlingFirst(n,k)*(k+1)^5: k in [0..n]]: n in [0..25]]; // Bruno Berselli, Mar 28 2013

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

a(n) = sum(k=0, n, stirling(n, k, 1)*(k+1)^5); \\ Michel Marcus, Nov 14 2015

a(n) = Sum_{k=0..n} Stirling1(n,k) * (k+1)^5.
From Seiichi Manyama, Apr 14 2025: (Start)
E.g.f.: Sum_{k>=0} (k+1)^5 * log(1+x)^k / k!.
E.g.f.: (1+x) * Sum_{k=0..5} Stirling2(6,k+1) * log(1+x)^k. (End)

A383049 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) is the n-th term of the inverse Stirling transform of j-> (j+1)^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 4, 1, 0, 1, 8, 5, -1, 0, 1, 16, 19, -3, 2, 0, 1, 32, 65, -1, 4, -6, 0, 1, 64, 211, 45, -10, -8, 24, 0, 1, 128, 665, 359, -116, 48, 20, -120, 0, 1, 256, 2059, 2037, -538, 340, -234, -52, 720, 0, 1, 512, 6305, 10079, -1316, 984, -1240, 1302, 72, -5040, 0

Offset: 0

Seiichi Manyama, Apr 14 2025

			Square array begins:
  1,  1,  1,    1,     1,     1,     1, ...
  1,  2,  4,    8,    16,    32,    64, ...
  0,  1,  5,   19,    65,   211,   665, ...
  0, -1, -3,   -1,    45,   359,  2037, ...
  0,  2,  4,  -10,  -116,  -538, -1316, ...
  0, -6, -8,   48,   340,   984, -1148, ...
  0, 24, 20, -234, -1240, -1866, 16400, ...

Christian G. Bower, PARI programs for transforms, 2007.
N. J. A. Sloane, Maple programs for transforms, 2001-2020.

Columns k=0..6 give A019590(n+1), A302190 (for n > 0), A222627, A222636, A222748, A223023, A383050.
Main diagonal gives A383051.
Cf. A362924, A362925.

a(n, k) = sum(j=0, n, (j+1)^k*stirling(n, j, 1));

A(n,k) = Sum_{j=0..n} (j+1)^k * Stirling1(n,j).
E.g.f. of column k: Sum_{j>=0} (j+1)^k * log(1+x)^j / j!.
E.g.f. of column k: (1+x) * Sum_{j=0..k} Stirling2(k+1,j+1) * log(1+x)^j.

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

A224095 Numerators of poly-Cauchy numbers c_n^(2).

Original entry on oeis.org

1, 1, -5, 11, -1103, 1627, -374473, 1220651, -92146157, 31595747, -20000218625, 176776749931, -5607610511548471, 374753409522157, -55207553310144173, 202183428095237231, -1614396705602979083803

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, A222627, A224094 (denominators).

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

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

A223173 Poly-Cauchy numbers c_3^(-n).

Original entry on oeis.org

-1, -3, -1, 45, 359, 2037, 10079, 46365, 204119, 873477, 3666959, 15191085, 62342279, 254119317, 1030760639, 4165958205, 16792710839, 67557739557, 271392171119, 1089053371725, 4366669645799, 17498051254197, 70086331418399, 280627721655645

Offset: 1

Takao Komatsu, Mar 28 2013

Definition of poly-Cauchy numbers in A222627.

Vincenzo Librandi, Table of n, a(n) for n = 1..300
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.
M. Z. Spivey, Combinatorial sums and finite differences, Discr. Math. 307 (24) (2007) 3130-3146
Wikipedia, Stirling transform
Index entries for linear recurrences with constant coefficients, signature (9,-26,24).

[&+[StirlingFirst(3,k)*(k+1)^n: k in [0..3]]: n in [1..25]]; // Bruno Berselli, Mar 28 2013

seq(2^(n+1)-3^(n+1)+4^n, n=0..30); # Robert Israel, Jun 21 2018

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

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

a(n) = Sum_{k=0..3} Stirling1(3,k)*(k+1)^n.
From Colin Barker, Mar 31 2013: (Start)
Conjecture:
a(n) = 2^(1+n) - 3^(1+n) + 4^n;
g.f.: -x*(6*x-1) / ((2*x-1)*(3*x-1)*(4*x-1)). (End)
Conjecture verified by Robert Israel, Jun 21 2018

A223852 Poly-Cauchy numbers c_5^(-n).

Original entry on oeis.org

-6, -8, 48, 340, 984, -1148, -34152, -254780, -1250376, -3417788, 12508248, 296104900, 3122953464, 26485493572, 201873508248, 1443404093380, 9892106472504, 65798800964932, 428187502981848, 2740792716574660, 17321987718906744, 108394003491348292

Offset: 1

Takao Komatsu, Mar 28 2013

Definition of poly-Cauchy numbers in A222627.

Vincenzo Librandi, Table of n, a(n) for n = 1..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.
M. Z. Spivey, Combinatorial sums and finite differences, Discr. Math. 307 (24) (2007) 3130-3146
Wikipedia, Stirling transform
Index entries for linear recurrences with constant coefficients, signature (20,-155,580,-1044,720).

[&+[StirlingFirst(5,k)*(k+1)^n: k in [0..5]]: n in [1..25]]; // Bruno Berselli, Mar 28 2013

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

a(n) = sum(k=0, 5, stirling(5, k, 1)*(k+1)^n); \\ Michel Marcus, Nov 14 2015

a(n) = Sum_{k=0..5} Stirling1(5,k)*(k+1)^n.

Empirical g.f.: -2*x*(810*x^3 - 361*x^2 + 56*x - 3) / ((2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)). - Colin Barker, Mar 31 2013

cp's OEIS Frontend

A223899 Poly-Cauchy numbers of the second kind hat c_n^(-2).

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A222636 Poly-Cauchy numbers c_n^(-3).

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A222748 Poly-Cauchy numbers c_n^(-4).

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A223023 Poly-Cauchy numbers c_n^(-5).

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A383049 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) is the n-th term of the inverse Stirling transform of j-> (j+1)^k.

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

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A224095 Numerators of poly-Cauchy numbers c_n^(2).

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PARI

A223173 Poly-Cauchy numbers c_3^(-n).

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