A075178 -id:A075178 - cp's OEIS frontend

A006232 Numerators of Cauchy numbers of first type.

1, 1, -1, 1, -19, 9, -863, 1375, -33953, 57281, -3250433, 1891755, -13695779093, 24466579093, -132282840127, 240208245823, -111956703448001, 4573423873125, -30342376302478019, 56310194579604163

Offset: 0

N. J. A. Sloane

The corresponding denominators are given in A006233.
-a(n+1), n>=0, also numerators from e.g.f. 1/x-1/log(1+x), with denominators A075178(n). |a(n+1)|, n>=0, numerators from e.g.f. 1/x+1/log(1-x) with denominators A075178(n). For formula of unsigned a(n) see A075178.
The signed rationals a(n)/A006233(n) provide the a-sequence for the Stirling2 Sheffer matrix A048993. See the W. Lang link concerning Sheffer a- and z-sequences.
Cauchy numbers of the first type are also called Bernoulli numbers of the second kind.
Named after the French mathematician, engineer and physicist Augustin-Louis Cauchy (1789-1857). - Amiram Eldar, Jun 17 2021

			1, 1/2, -1/6, 1/4, -19/30, 9/4, -863/84, 1375/24, -33953/90, ...

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
Harold Jeffreys and B. S. Jeffreys, Methods of Mathematical Physics, Cambridge, 1946, p. 259.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..100
Arnold Adelberg, 2-adic congruences of Norland numbers and of Bernoulli numbers of the second kind, J. Number Theory, Vol. 73, No. 1 (1998), pp. 47-58.
I. S. Gradsteyn and I. M. Ryzhik, Table of integrals, series and products, (1980), page 2 (formula 0.131).
L. B. W. Jolley, Summation of Series, Dover, (1961) (formula 70).
Wolfdieter Lang, Sheffer a- and z-sequences.
Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
Hong-Mei Liu, Shu-Hua Qi and Shu-Yan Ding, Some Recurrence Relations for Cauchy Numbers of the First Kind, JIS, Vol. 13 (2010), Article 10.3.8.
Rui-Li Liu and Feng-Zhen Zhao, Log-concavity of two sequences related to Cauchy numbers of two kinds, Online Journal of Analytic Combinatorics, Issue 14 (2019), #09.
Donatella Merlini, Renzo Sprugnoli and M. Cecilia Verri, The Cauchy numbers, Discrete Math., Vol. 306, No. 16 (2006), pp. 1906-1920.
Eric Weisstein's World of Mathematics, Bernoulli Numbers of the Second Kind.
Ming Wu and Hao Pan, Sums of products of Bernoulli numbers of the second kind, Fib. Quart., Vol. 45, No. 2 (2007), pp. 146-150.
Feng-Zhen Zhao, Sums of products of Cauchy numbers, Discrete Math., Vol. 309, No. 12 (2009), pp. 3830-3842.

Cf. A006233 (denominators), A002206, A002207, A002208, A002209, A002657, A002790.

[Numerator((&+[StirlingFirst(n,k)/(k+1): k in [0..n]])): n in [0..20]]; // G. C. Greubel, Nov 13 2018

seq(numer(add(stirling1(n,k)/(k+1),k=0..n)),n=0..20); # Peter Luschny, Apr 28 2009

a[n_] := Numerator[ Sum[ StirlingS1[n, k]/(k + 1), {k, 0, n}]]; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Nov 03 2011, after Maple *)
a[n_] := Numerator[ Integrate[ Gamma[x+1]/Gamma[x-n+1], {x, 0, 1}]]; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Jul 29 2013 *)
a[ n_] := If[ n < 0, 0, (-1)^n Numerator @ Integrate[ Pochhammer[ -x, n], {x, 0, 1}]]; (* Michael Somos, Jul 12 2014 *)
a[ n_] := If[ n < 0, 0, Numerator [ n! SeriesCoefficient[ x / Log[ 1 + x], {x, 0, n}]]]; (* Michael Somos, Jul 12 2014 *)
Join[{1}, Array[Numerator[(1/#) Integrate[Product[(x - k), {k, 0, # - 1}], {x, 0, 1}]] &, 25]] (* Michael De Vlieger, Nov 13 2018 *)

for(n=0,20, print1(numerator( sum(k=0,n, stirling(n, k, 1)/(k+1)) ), ", ")) \\ G. C. Greubel, Nov 13 2018

# Results are abs values
from fractions import gcd
aa,n,sden = [0,1],1,1
while n < 20:
    j,snom,sden,a = 1,0,(n+1)*sden,0
    while j < len(aa):
        snom,j = snom+aa[j]*(sden//(j+1)),j+1
    nom,den = snom,sden
    print(n,nom//gcd(nom,den))
    aa,j = aa+[-aa[j-1]],j-1
    while j > 0:
        aa[j],j = n*aa[j]-aa[j-1],j-1
    n = n+1 # A.H.M. Smeets, Nov 14 2018

from fractions import Fraction
from sympy.functions.combinatorial.numbers import stirling
def A006232(n): return sum(Fraction(stirling(n,k,kind=1,signed=True),k+1) for k in range(n+1)).numerator # Chai Wah Wu, Jul 09 2023

def A006232_list(len):
    f, R, C = 1, [1], [1]+[0]*(len-1)
    for n in (1..len-1):
        for k in range(n, 0, -1):
            C[k] = -C[k-1] * k / (k + 1)
        C[0] = -sum(C[k] for k in (1..n))
        R.append((C[0]*f).numerator())
        f *= n
    return R
print(A006232_list(20)) # Peter Luschny, Feb 19 2016

Numerator of integral of x(x-1)...(x-n+1) from 0 to 1.
E.g.f.: x/log(1+x). (Note: the numerator of the coefficient of x^n/n! is a(n) - Michael Somos, Jul 12 2014)
Numerator of Sum_{k=0..n} A048994(n,k)/(k+1). - Peter Luschny, Apr 28 2009
Sum_{k=1..n} 1/k = C + log(n) + 1/(2n) + Sum_{k=2..inf} |a(n)|/A075178(n-1) * 1/(n*(n+1)*...*(n+k-1)) (section 0.131 in Gradshteyn and Ryzhik tables). - Ralf Stephan, Jul 12 2014
a(n) = numerator(f(n) * n!), where f(0) = 1, f(n) = Sum_{k=0..n-1} (-1)^(n-k+1) * f(k) / (n-k+1). - Daniel Suteu, Feb 23 2018
Sum_{k = 1..n} (1/k) = A001620 + log(n) + 1/(2n) - Sum_{k >= 2} abs((a(k)/A006233(k)/k/(Product_{j = 0..k-1} (n-j)))), (see I. S. Gradsteyn, I. M. Ryzhik). - A.H.M. Smeets, Nov 14 2018

A075181 Coefficients of certain polynomials (rising powers).

Original entry on oeis.org

1, 2, 1, 6, 6, 2, 24, 36, 22, 6, 120, 240, 210, 100, 24, 720, 1800, 2040, 1350, 548, 120, 5040, 15120, 21000, 17640, 9744, 3528, 720, 40320, 141120, 231840, 235200, 162456, 78792, 26136, 5040, 362880, 1451520, 2751840, 3265920, 2693880, 1614816

Offset: 1

Wolfdieter Lang, Sep 19 2002

This is the unsigned triangle A048594 with rows read backwards.
The row polynomials p(n,y) := Sum_{m=0..n-1}a(n,m)*y^m, n>=1, are obtained from (log(x)*(-x*log(x))^n)*(d^n/dx^n)(1/log(x)), n>=1, after replacement of log(x) by y.
The gcd of row n is A075182(n). Row sums give A007840(n), n>=1.
The columns give A000142 (factorials), A001286 (Lah), 2* A075183, 2*A075184, 4*A075185, 4!*A075186, 4!*A075187 for m=0..6.
Coefficients T(n,k) of the differential operator expansion
[x^(1+y)D]^n = x^(n*y)[T(n,1)* (xD)^n / n! + y * T(n,2)* (xD)^(n-1) / (n-1)! + ... + y^(n-1) * T(n,n) * (xD)], where D = d/dx. Note that (xD)^n = Bell(n,:xD:), where (:xD:)^n = x^n * D^n and Bell(n,x) are the Bell / Touchard polynomials. See A094638. - Tom Copeland, Aug 22 2015

			Triangle starts:
1;
2,1;
6,6,2;
24,36,22,6;
...
n=2: (x^2*log(x)^3)*(d^2/d^x^2)(1/log(x)) = 2 + log(x).

Vincenzo Librandi, Rows n = 1..100, flattened
Y.-Z. Huang, J. Lepowsky and L. Zhang, A logarithmic generalization of tensor product theory for modules for a vertex operator algebra, arXiv:math/0311235 [math.QA], 2003; Internat. J. Math. 17 (2006), no. 8, 975-1012. See page 984 eq. (3.9) MR2261644.
D. Lubell, Problem 10992, problems and solutions, Amer. Math. Monthly 110 (2003) p. 155. Equal Sums of Reciprocal Products: 10992 (2004) pp. 827-829.

Cf. A048594, A075178, A007840, A075182.

Cf. A094638.

seq(seq(k!*abs(Stirling1(n,k)),k=n..1,-1),n=1..10); # Robert Israel, Jul 12 2015

Table[ Table[ k!*StirlingS1[n, k] // Abs, {k, 1, n}] // Reverse, {n, 1, 9}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)

{T(n, k)= if(k<0 || k>=n, 0, (-1)^k* stirling(n, n-k)* (n-k)!)} /* Michael Somos Apr 11 2007 */

a(n, m) = (n-m)!*|S1(n, n-m)|, n>=m+1>=1, else 0, with S1(n, m) := A008275(n, m) (Stirling1).

a(n, m) = (n-m)*a(n-1, m)+(n-1)*a(n-1, m-1), if n>=m+1>=1, a(n, -1) := 0 and a(1, 0)=1, else 0.

A321331 Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S2 = A048993 (Stirling2).

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 14, 18, 4, 1, 30, 75, 40, 5, 1, 62, 270, 260, 75, 6, 1, 126, 903, 1400, 700, 126, 7, 1, 254, 2898, 6804, 5250, 1596, 196, 8, 1, 510, 9075, 31080, 34755, 15876, 3234, 288, 9, 1, 1022, 27990, 136420, 212625, 136962, 41160, 6000, 405, 10, 1, 2046, 85503, 583000, 1233650, 1076922, 447909, 95040, 10395, 550, 11

Offset: 0

Wolfdieter Lang, Dec 03 2018

This lower triangular matrix T is the inverse of the triangular matrix with elements Narumi[-1](n,m)/(m+1) = S1(n+1, m+1)/(n+1), with the Narumi triangle for parameter a = -1, and S1 = A048994 (Stirling1), i.e., Sum_{k=m..n} T(n, k) * S1(k+1, m+1)/(k+1) = delta_{n,m} (Kronecker symbol).
This triangle arises from the inverse of the rational Sheffer matrix Narumi[-1] = (log(1+x)/x, log(1+x) (such special Sheffer matrices (g(x), x*g(x)) define elements of the Narumi subgroup). The inverse matrix is (Narumi[-1])^(-1) = ((exp(x) - 1)/x, exp(x) - 1).
In order to have an integer matrix one takes T(n, k) := (n+1)*(Narumi[-1])^(-1)(n, k) = (k+1)*S2(n+1, k+1). The connection to S2 = A048993 results from the general relation between each Narumi-type matrix N = (g(x), x*g(x)) and its associated Sheffer matrix J = (1, x*g(x)) (this is of the Jabotinsky-type), i.e., N(n, m) = (m+1)*J(n+1, m+1)/(n+1), or with the row polynomials Npol(n, x) = (1/(n+1))*(d/dx)Jpol(n+1, x).
The signed triangle (-1)^(n-k)*A028421(n, k) (with upper diagonals filled with zeros) gives the integer matrix Nscaled with elements (n+1)*Narumi[-1](n,k). This inverse of Nscaled has the rational elements (Narumi[-1])^(-1)(k, m)/(m+1) = (1/(k+1))*S2(k+1, m+1).
The a- and z- sequence for the Sheffer matrix (Narumi[-1])^(-1) (see A006232 for a link on these sequences) have e.g.f.s Ea(x) = x/log(1 + x) and Ez(x) = 1/log(1 + x) - 1/x, hence a(n) = A006232(n)/A006233(n) and z(n) = A006232(n+1)/A075178(n), for n >= 0. This leads to the recurrence for T(n, k) given in the formula section.
The Boas-Buck-type column recurrence (see the link, also for references) uses the sequence with o.g.f. GBB(y) = exp(y)/(exp(y) - 1) - 1/y, with BB(n) = (-1)^(n+1)*A060054(n+1 ) / A227830(n+1), for n >= 1. For the recurrence see the formula section.
The Meixner-type identity (see the Meixner link) for the row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k, derived from the one for the Narumi[-1]^(-1) row polynomials is Sum_{k=1..n} (-1)^{k+1}*(1/k)*(d/dx)^k R(n, x)/(n+1) = R(n-1, x), for n >= 1, and R(0, x) = 1. Here d/dx is a differentiation operator.
The Roman-type recurrence for the row polynomials (see the reference, Corollary 3.7.2. p. 50) becomes, with the z-sequence from above: R(n, x) = ((n+1)/n)*{(x + 1/2)*1 + (x - z(1))*d/dx - Sum_{k=2..n-1} (1/k!)*z(k)*(d/dx)^k}*R(n-1, x), for n >= 1, and R(0, x) = 1.
The triangle is the exponential Riordan square (cf. A321620) of exp(x)-1 with an additional main diagonal of zeros. - Peter Luschny, Jan 03 2019

			The triangle T(n, k) begins:
  n\k  0    1     2      3       4       5      6     7     8   9 10 ...
  ----------------------------------------------------------------------
  0:   1
  1:   1    2
  2:   1    6     3
  3:   1   14    18      4
  4:   1   30    75     40       5
  5:   1   62   270    260      75       6
  6:   1  126   903   1400     700     126      7
  7:   1  254  2898   6804    5250    1596    196     8
  8:   1  510  9075  31080   34755   15876   3234   288     9
  9:   1 1022 27990 136420  212625  136962  41160  6000   405  10
  10:  1 2046 85503 583000 1233650 1076922 447909 95040 10395 550 11
  ...
Recurrence (from Stirling2): T(4, 2) = 3*(T(3, 2) + T(3, 1)/2) = 3*(18 + 14/2) = 75.
Recurrence (from a- and z-sequence): T(4, 0) = 5*((1/2)*T(3, 0) - (1/12)*T(3, 1) + (1/12)*T(3, 2) - (19/120)*T(3, 3)) = 5*(1/2 - 14/12 + 18/12 - 4*19/120) = 1; T(4,2) = (5/2)*(1*1*T(3, 1) + 2*(1/2)*T(3, 2) + 3*(-1/6)* T(3, 3)) = (5/2)*(14 + 18 - 2) = 75.
Recurrence for column k=2 (Boas-Buck-type): T(4, 2) = (5!*3/2)*((1/3!)*(1/12)*T(2, 2) + (1/4!)*(1/2)*T(3, 2)) = (5!*3/2)*((1/72)*3 + (1/48)*18) = 75.
Meixner identity for the row polynomials, for n = 3: {d/dx  - (1/2)*(d/dx)^2 + (1/3)*(d/dx)^3)}*R(3, x)/4) = ((14 - 36/2 + 24/3) + (36 - 24/2)*x + 12*x^2)/4 = (1 + 6*x + 3*x^2) = R(2, x).
Roman type recurrence for row polynomials: R(n, 3) = (3/2)*{(x + 1/12)*(1 + 6*x + 3*x^2) + (x - (-1/2))*(6 + 6*x) - (1/2!)*(1/12)*6} = 1 + 14*x + 18*x^2 + 4*x^3.

Steven Roman, The umbral calculus, Academic Press, 1984.

Wolfdieter Lang, On the Generating Functions of the Boas-Buck Sequences for the Inverse of Riordan and Sheffer Matrices
J. Meixner, Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion, J. Lond. Math. Soc. 9 (1934), 6-13.

Cf. A006232, A006233, A060054, A028421, A048993, A048994, A075178, A227830.

Flat(List([0..10],n->List([0..n],k->(k+1)*Stirling2(n+1,k+1)))); # Muniru A Asiru, Dec 03 2018

T:=(n,k)->(k+1)*Stirling2(n+1,k+1): seq(seq(T(n,k),k=0..n),n=0..10); # Muniru A Asiru, Dec 03 2018

T[n_, k_] := (k+1) * StirlingS2[n+1, k+1];  Table[T[n, k], {n,0,10}, {k, 0, n}] //Flatten (* Amiram Eldar, Dec 03 2018 *)

T(n, k) = (k+1)*stirling(n+1, k+1, 2) \\ Thomas Scheuerle, Nov 10 2023

# uses[riordan_square from A321620]
riordan_square(exp(x) - 1, 10, True) # Peter Luschny, Jan 03 2019

T(n, k) = (k+1)*A048993(n+1, k+1), with A048993 = Stirling2, for n >= k >= 0, and 0 otherwise.
T(n, k) = (n+1)*(Narumi[a=-1])^(-1)(n, k), with the Narumi[a=-1] matrix with entries (-1)^(n-k)*A028421(n, k)/(n+1).
E.g.f. for column k sequence: E(k, x) = (x*d/dx + 1)*EN(k, x), where EN(k, x) = (exp(x) - 1)^(k+1)/(x*k!) is the e.g.f. for the (Narumi[a=-1])^(-1) columns. Hence E(k, x) = exp(x)*(exp(x) - 1)*(k+1)/k!, for k >= 0.
E.g.f. for (ordinary) row polynomials R(n, x): Epol(z, x) = exp(z)*exp(x*(exp(z) - 1))*(1 + x*(exp(z) - 1)).
Recurrence (from Stirling2): T(n, k) = 0 for n < k; T(n, 0) = (k + 1)*T(n-1, k), for n <= 1, T(0, 0) = 1; T(n, k) = (k+1)*(T(n-1, k) + T(n-1, k-1)/k), for n >= 1, k >= 1.
Recurrence (from a- and z-sequence, see above): a = {1, 1/2, -1/6, 1/4, -19/30, 9/4, ...}, z = {1/2, -1/12, 1/12, -19/120, 9/20, -863/504, ...}.
T(n, k) = 0, for n < k; T(n, 0) = (n+1)*Sum_{j=0..n-1} z(j)*T(n-1, j), for n >= 1, with T(0, 0) = 1; T(n, k) = ((n+1)/k)*Sum_{j=0..n-m} binomial(k-1+j, j)*a(j)*T(n-1, k-1+j).
Recurrence for column k, from the Boas-Buck-type sequence BB(n) = (-1)^(n+1)*A060054(n+1)/A227830(n+1), for n >= 0; BB = {1/2, 1/12, 0, -1/720, 0, 1/30240, 0, -1/1209600, ...}: T(n, k) = 0, for  n < k; T(n, n) = n+1, for n >= 0; T(n, k) = ((n+1)!*(k+1)/(n-k))*Sum_{j=k..n-1} (1/(j+1)!)*BB(n-(j+1))*T(j, k), for n >= 0 and k = 0, 1, ..., n-1.
T(n, k) = Stirling2(n+2, k+1) - Stirling2(n+1, k). - Peter Luschny, May 26 2020

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A075179 One half of A075178.

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A006232 Numerators of Cauchy numbers of first type.

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A075181 Coefficients of certain polynomials (rising powers).

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A321331 Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S2 = A048993 (Stirling2).

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A110841 a(n) is the number of prime factors, with multiplicity, of abs(A014509(n)).

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