A223899 -id:A223899 - cp's OEIS frontend

A223901 Poly-Cauchy numbers of the second kind hat c_n^(-3).

1, -8, 35, -161, 854, -5248, 36966, -294714, 2628600, -25963392, 281529192, -3326287848, 42546905712, -585889457328, 8643254959008, -136013600978784, 2274436197944064, -40278639752011008, 753115809287568384, -14826614346669090816, 306574242780102220800

Offset: 0

Takao Komatsu, Mar 29 2013

sign

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 with a q parameter, Ramanujan J. 31 (2013), 353-371.
Takao Komatsu, Poly-Cauchy numbers, RIMS Kokyuroku 1806 (2012).
Takao Komatsu, Poly-Cauchy numbers, Kyushu J. Math. 67 (2013), 143-153.

Cf. A081046, A223899, A223902, A223904.

Cf. A222636.

[&+[StirlingFirst(n, k)*(-1)^k*(k+1)^3: k in [0..n]]: n in [0..25]];

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

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

a(n) = Sum_{k=0..n} Stirling1(n,k) * (-1)^k * (k+1)^3.
E.g.f.: (1 - 7 * log(1 + x) + 6 * log(1 + x)^2 - log(1 + x)^3) / (1 + x). - Ilya Gutkovskiy, Aug 10 2021
From Seiichi Manyama, Apr 15 2025: (Start)
E.g.f.: Sum_{k>=0} (k+1)^3 * (-log(1+x))^k / k!.
a(n) = (-1)^n * Sum_{k=0..3} k! * Stirling2(4,k+1) * |Stirling1(n+1,k+1)|. (End)

A223902 Poly-Cauchy numbers of the second kind hat c_n^(-4).

Original entry on oeis.org

1, -16, 97, -531, 3148, -20940, 156680, -1310840, 12166096, -124281120, 1387313520, -16813355280, 219967479744, -3090914335104, 46439677053120, -743069262651840, 12616998421804416, -226608929801923968, 4292762009479969536, -85545808260446050560, 1789078468694176410624

Offset: 0

Takao Komatsu, Mar 29 2013

sign

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 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, A223899, A223901, A223904.

Cf. A222748.

Table[Sum[StirlingS1[n, k] (-1)^k (k + 1)^4, {k, 0, n}], {n, 0, 30}]

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

a(n) = Sum_{k=0..n} (-1)^k * (k+1)^4 * Stirling1(n,k).
From Seiichi Manyama, Apr 15 2025: (Start)
E.g.f.: Sum_{k>=0} (k+1)^4 * (-log(1+x))^k / k!.
E.g.f.: (1/(1+x)) * Sum_{k=0..4} Stirling2(5,k+1) * (-log(1+x))^k.
a(n) = (-1)^n * Sum_{k=0..4} k! * Stirling2(5,k+1) * |Stirling1(n+1,k+1)|. (End)

A223904 Poly-Cauchy numbers of the second kind hat c_n^(-5).

Original entry on oeis.org

1, -32, 275, -1817, 12134, -87784, 699894, -6158058, 59566464, -630057696, 7246806720, -90151868160, 1207028135520, -17314992935040, 265048030579680, -4313510679824160, 74387763047472000, -1355291635314213120, 26016022725597866880, -524865277479851360640, 11103724030717930095360

Offset: 0

Takao Komatsu, Mar 29 2013

sign

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 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, A223899, A223901, A223902.

Cf. A223023.

[&+[StirlingFirst(n, k)*(-1)^k*(k+1)^5: k in [0..n]]: n in [0..23]]; // Vincenzo Librandi, Aug 03 2013

Table[Sum[StirlingS1[n, k] (-1)^k (k + 1)^5, {k, 0, n}], {n, 0, 30}]

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

a(n) = Sum_{k=0..n} (-1)^k * (k+1)^5 * Stirling1(n,k).
From Seiichi Manyama, Apr 15 2025: (Start)
E.g.f.: Sum_{k>=0} (k+1)^5 * (-log(1+x))^k / k!.
E.g.f.: (1/(1+x)) * Sum_{k=0..5} Stirling2(6,k+1) * (-log(1+x))^k.
a(n) = (-1)^n * Sum_{k=0..5} k! * Stirling2(6,k+1) * |Stirling1(n+1,k+1)|. (End)

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

Original entry on oeis.org

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

A224102 Numerators of poly-Cauchy numbers of the second kind hat c_n^(2).

Original entry on oeis.org

1, -1, 13, -43, 5647, -3401, 2763977, -10326059, 876576493, -1665984623, 1156096889861, -2220482068331, 75970695882225719, -1088498788093641, 855021689397409453, -3324381371618385007, 4010325276269988793421

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, A223899, A219247 (denominators).

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

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

A344639 Array read by ascending antidiagonals: A(n, k) is the number of (n, k)-poly-Cauchy permutations.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 5, 4, 1, 24, 17, 13, 8, 1, 120, 74, 51, 35, 16, 1, 720, 394, 244, 161, 97, 32, 1, 5040, 2484, 1392, 854, 531, 275, 64, 1, 40320, 18108, 9260, 5248, 3148, 1817, 793, 128, 1, 362880, 149904, 70508, 36966, 20940, 12134, 6411, 2315, 256, 1

Offset: 0

Stefano Spezia, May 25 2021

An (n, k)-poly-Cauchy permutation is a permutation which satisfies the properties listed by Bényi and Ramírez in Definition 1.

			n\k|   0     1     2     3     4 ...
---+----------------------------
0  |   1     1     1     1     1 ...
1  |   1     2     4     8    16 ...
2  |   2     5    13    35    97 ...
3  |   6    17    51   161   531 ...
4  |  24    74   244   854  3148 ...
...

Beáta Bényi and José Luis Ramírez, Poly-Cauchy numbers -- the combinatorics behind, arXiv:2105.04791 [math.CO], 2021.

Rows n=0..2 give A000012, A000079, A007689.
Columns k=0..5 give A000142, A000774, |A223899|, |A223901|, |A223902|, |A223904|.
Main diagonal gives A192563.
Antidiagonal sums give A344640.
Cf. A007318, A008275, A008277, A081048.

A[n_,k_]:=Sum[Abs[StirlingS1[n,m]](m+1)^k,{m,0,n}]; Flatten[Table[A[n-k,k],{n,0,9},{k,0,n}]]

A(n, k) = Sum_{m=0..n} abs(S1(n, m)) * (m + 1)^k, where S1 indicates the signed Stirling numbers of first kind (see Theorem 5 in Bényi and Ramírez).
A(n, 0) = n! = A000142(n) (see Example 6 in Bényi and Ramírez).
A(1, k) = 2^k = A000079(k) (see Example 7 in Bényi and Ramírez).
A(2, k) = 2^k + 3^k = A007689(k) (see Example 8 in Bényi and Ramírez).
Sum_{m=0..n} (-1)^m*S2(n, m)*A(m, k) = (-1)^n*(n + 1)^k, where S2 indicates the Stirling numbers of the second kind (see Theorem 9 in Bényi and Ramírez).
A(n, k) = Sum_{j=0..k} j! * abs(S1(n+1, j+1)) * S2(k+1, j+1) (see Theorem 14 in Bényi and Ramírez).
A(n, k) = (n - 1)*A(n-1, k) + Sum_{i=0..k} C(k, i)*A(n-1, k-i) for n > 0 (see Theorem 15 in Bényi and Ramírez).
A(n, k) = Sum_{i=0..n} Sum_{j=0..k} C(n-1, i)*i!*C(k, j)*A(n-1-i, k-j) for n > 0 (see Theorem 17 in Bényi and Ramírez).
A(n, k) = Sum_{m=0..n} Sum_{i=0..m} C(k-i, m-i)*S2(k, i)*abs(S1(n+1, m+1)) (see Theorem 18 in Bényi and Ramírez).
From Seiichi Manyama, Apr 15 2025: (Start)
E.g.f. of column k: Sum_{j>=0} (j+1)^k * (-log(1-x))^j / j!.
E.g.f. of column k: (1/(1-x)) * Sum_{j=0..k} Stirling2(k+1,j+1) * (-log(1-x))^j. (End)

cp's OEIS Frontend

A223901 Poly-Cauchy numbers of the second kind hat c_n^(-3).

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A223902 Poly-Cauchy numbers of the second kind hat c_n^(-4).

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A223904 Poly-Cauchy numbers of the second kind hat c_n^(-5).

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

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A224102 Numerators of poly-Cauchy numbers of the second kind hat c_n^(2).

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PARI

A344639 Array read by ascending antidiagonals: A(n, k) is the number of (n, k)-poly-Cauchy permutations.

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