A383049 -id:A383049 - cp's OEIS frontend

A222627 Poly-Cauchy numbers c_n^(-2) (for definition see Comments lines).

1, 4, 5, -3, 4, -8, 20, -52, 72, 936, -17568, 238752, -3113280, 41503680, -577877760, 8470414080, -131039838720, 2139954163200, -36854615347200, 668374040678400, -12742107588403200, 254904791591116800, -5341386032640000000, 117034910701793280000

Offset: 0

Takao Komatsu, Mar 28 2013

sign

The definition of poly-Cauchy numbers is given in Theorem 1 of the paper Poly-Cauchy numbers (see Links lines).

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..200
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=2 of A383049.

Cf. A006233.

[&+[StirlingFirst(n,k)*(k+1)^2: k in [0..n]]: n in [0..25]]; // Bruno Berselli, Mar 28 2013

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

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

a(n) = Sum_{k=0..n} Stirling1(n,k) * (k+1)^2.
E.g.f.: (1 + x) * (1 + log(1 + x) * (3 + log(1 + x))). - Ilya Gutkovskiy, Aug 09 2021
E.g.f.: Sum_{k>=0} (k+1)^2 * log(1+x)^k / k!. - Seiichi Manyama, Apr 14 2025

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)

A383050 a(n) = Sum_{k=0..n} (k+1)^6 * Stirling1(n,k).

Original entry on oeis.org

1, 64, 665, 2037, -1316, -1148, 16400, -116032, 809592, -6059424, 49512792, -442266888, 4302605280, -45351578400, 515054655360, -6268075470720, 81309027784320, -1118525784929280, 16235659302272640, -247395991797912960, 3936073920965890560, -64988868076072657920

Offset: 0

Seiichi Manyama, Apr 14 2025

sign

Inverse Stirling transform of (n+1)^6.

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

Column k=6 of A383049.

a(n) = sum(k=0, n, (k+1)^6*stirling(n, k, 1));

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^6*log(1+x)^k/k!)))

E.g.f.: Sum_{k>=0} (k+1)^6 * log(1+x)^k / k!.

E.g.f.: (1+x) * Sum_{k=0..6} Stirling2(7,k+1) * log(1+x)^k.

A383051 a(n) is the n-th term of the inverse Stirling transform of j-> (j+1)^n.

Original entry on oeis.org

1, 2, 5, -1, -116, 984, 16400, -788418, 5474016, 941115360, -51647682648, -264087895512, 244846563852864, -16953959408998080, -436871956049596800, 219647419965976413744, -20283048895473275917824, -877465277974899660349440, 545297904370739513319183360

Offset: 0

Seiichi Manyama, Apr 14 2025

sign

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

Main diagonal of A383049.

Cf. A320082, A367820.

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

a(n) = Sum_{k=0..n} (k+1)^n * Stirling1(n,k).
a(n) = n! * [x^n] Sum_{k>=0} (k+1)^n * log(1+x)^k / k!.
a(n) = n! * [x^n] (1+x) * Sum_{k=0..n} Stirling2(n+1,k+1) * log(1+x)^k.

cp's OEIS Frontend

A222627 Poly-Cauchy numbers c_n^(-2) (for definition see Comments lines).

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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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A383050 a(n) = Sum_{k=0..n} (k+1)^6 * Stirling1(n,k).

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PARI

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A383051 a(n) is the n-th term of the inverse Stirling transform of j-> (j+1)^n.

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Programs

PARI

Formula