A222636 -id:A222636 - 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)

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.

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

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

A383052 a(n) = Sum_{k=0..n} (k+1)^3 * Stirling2(n,k).

Original entry on oeis.org

1, 8, 35, 153, 706, 3479, 18313, 102678, 610989, 3844525, 25492752, 177579961, 1295811637, 9879799744, 78525094847, 649253421173, 5573667453498, 49595062947091, 456689512735421, 4345710521536150, 42675672248378721, 431963852263306569, 4501627598926298992

Offset: 0

Seiichi Manyama, Apr 14 2025

nonn

Stirling transform of (n+1)^3.

Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A000110, A222636, A362925.

a(n) = sum(k=0, n, (k+1)^3*stirling(n, k, 2));

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^3*(exp(x)-1)^k/k!)))

a(n) = A362925(n+3,3).
E.g.f.: Sum_{k>=0} (k+1)^3 * (exp(x) - 1)^k / k!.
E.g.f.: exp(exp(x) - 1) * Sum_{k=0..3} Stirling2(4,k+1) * (exp(x) - 1)^k.

cp's OEIS Frontend

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

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

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

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A383052 a(n) = Sum_{k=0..n} (k+1)^3 * Stirling2(n,k).

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