A223023 -id:A223023 - cp's OEIS frontend

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

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)

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.

A224100 Denominators of poly-Cauchy numbers c_n^(5).

Original entry on oeis.org

1, 32, 7776, 82944, 388800000, 51840000, 2613824640000, 11948912640000, 3629482214400000, 806551603200000, 77937565348177920000, 14170466426941440000, 92074412343521441433600000

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, A223023, A224094, A224096, A224098, A224101 (numerators).

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

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

A224101 Numerators of poly-Cauchy numbers c_n^(5).

Original entry on oeis.org

1, 1, -211, 4241, -57453709, 29825987, -7362684132917, 198504470798947, -415989828245529323, 730328251215062341, -628191544925589374756597, 1131010588175721446183783, -80125844020238574218022657310343

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..250
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. A006232, A223023, A224095, A224097, A224099, A224100 (denominators).

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

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

A383054 a(n) = Sum_{k=0..n} (k+1)^5 * Stirling2(n,k).

Original entry on oeis.org

1, 32, 275, 1785, 11002, 68303, 436297, 2891670, 19947717, 143327725, 1072207680, 8342947657, 67440657877, 565603592392, 4914839764895, 44191989524117, 410644596021954, 3938713285932859, 38950532224469117, 396712750010963782, 4157217331880368521

Offset: 0

Seiichi Manyama, Apr 14 2025

nonn

Stirling transform of (n+1)^5.

Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A000110, A223023, A362925.

a(n) = sum(k=0, n, (k+1)^5*stirling(n, k, 2));

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^5*(exp(x)-1)^k/k!)))

a(n) = A362925(n+5,5).
E.g.f.: Sum_{k>=0} (k+1)^5 * (exp(x) - 1)^k / k!.
E.g.f.: exp(exp(x) - 1) * Sum_{k=0..5} Stirling2(6,k+1) * (exp(x) - 1)^k.

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A223904 Poly-Cauchy numbers of the second kind hat 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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A224100 Denominators of poly-Cauchy numbers c_n^(5).

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A224101 Numerators of poly-Cauchy numbers c_n^(5).

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A383054 a(n) = Sum_{k=0..n} (k+1)^5 * Stirling2(n,k).

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