A007840 -id:A007840 - cp's OEIS frontend

A006252 Expansion of e.g.f. 1/(1 - log(1+x)).

1, 1, 1, 2, 4, 14, 38, 216, 600, 6240, 9552, 319296, -519312, 28108560, -176474352, 3998454144, -43985078784, 837126163584, -12437000028288, 237195036797184, -4235955315745536, 85886259443020800, -1746536474655406080, 38320721602434017280, -864056965711935974400

Offset: 0

N. J. A. Sloane

sign

From Michael Somos, Mar 04 2004: (Start)
Stirling transform of a(n+1)=[1,2,4,14,38,...] is A000255(n)=[1,3,11,53,309,...].
Stirling transform of 2*a(n)=[2,2,4,8,28,...] is A052849(n)=[2,4,12,48,240,...].
Stirling transform of a(n)=[1,1,2,4,14,38,216,...] is A000142(n)=[1,2,6,24,120,...].
Stirling transform of a(n-1)=[1,1,1,2,4,14,38,...] is A000522(n-1)=[1,2,5,16,65,...].
Stirling transform of a(n-1)=[0,1,1,2,4,14,38,...] is A007526(n-1)=[0,1,4,15,64,...].
(End)
For n > 0: a(n) = sum of n-th row in triangle A048594. - Reinhard Zumkeller, Mar 02 2014
Coefficients in a factorial series representation of the exponential integral: exp(z)*E_1(z) = Sum_{n >= 0} (-1)^n*a(n)/(z)n, where (z)_n denotes the rising factorial z*(z + 1)*...*(z + n) and E_1(z) = Integrate{t = z..inf} exp(-t)/t dt. See Weninger, equation 6.4. - Peter Bala, Feb 12 2019

G. Pólya, Induction and Analogy in Mathematics. Princeton Univ. Press, 1954, p. 9.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
Beáta Bényi and Daniel Yaqubi, Mixed coloured permutations, arXiv:1903.07450 [math.CO], 2019.
Takao Komatsu and Amalia Pizarro-Madariaga, Harmonic numbers associated with inversion numbers in terms of determinants, Turkish Journal of Mathematics (2019) Vol. 43, 340-354.
E. J. Weniger, Summation of divergent power series by means of factorial series arXiv:1005.0466v1 [math.NA], 2010.

Column k=1 of A320080.

Cf. A007840.

a006252 0 = 1
a006252 n = sum $ a048594_row n  -- Reinhard Zumkeller, Mar 02 2014

With[{nn=30},CoefficientList[Series[1/(1-Log[1+x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 12 2016 *)

a(n)=if(n<0,0,n!*polcoeff(1/(1-log(1+x+x*O(x^n))),n))

{a(n)=local(CF=1+x*O(x^n)); for(k=0, n-1, CF=1/((n-k+1)-(n-k)*x+(n-k+1)^2*x*CF)); n!*polcoeff(1+x/(1-x+x*CF), n, x)} /* Paul D. Hanna, Dec 31 2011 */

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (-1)^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v; \\ Seiichi Manyama, May 22 2022

def A006252_list(len):
    f, R, C = 1, [1], [1]+[0]*len
    for n in (1..len):
        f *= n
        for k in range(n, 0, -1):
            C[k] = -C[k-1]*((k-1)/(k) if k>1 else 1)
        C[0] = -sum(C[k] for k in (1..n))
        R.append(C[0]*f)
    return R
print(A006252_list(24)) # Peter Luschny, Feb 21 2016

a(n) = Sum_{k=0..n} k!*stirling1(n, k). - Vladeta Jovovic, Sep 08 2002
a(n) = D^n(1/(1-x)) evaluated at x = 0, where D is the operator exp(-x)*d/dx. Row sums of A048594. Cf. A007840. - Peter Bala, Nov 25 2011
E.g.f.: 1/(1-log(1+x)) = 1 + x/(1-x + x/(2-x + 4*x/(3-2*x + 9*x/(4-3*x + 16*x/(5-4*x + 25*x/(6-5*x +...)))))), a continued fraction. - Paul D. Hanna, Dec 31 2011
a(n)/n! ~ -(-1)^n / (n * (log(n))^2) * (1 - 2*(1 + gamma)/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 01 2018
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, May 22 2022

A003713 Expansion of e.g.f. log(1/(1+log(1-x))).

Original entry on oeis.org

0, 1, 2, 7, 35, 228, 1834, 17582, 195866, 2487832, 35499576, 562356672, 9794156448, 186025364016, 3826961710272, 84775065603888, 2011929826983504, 50929108873336320, 1369732445916318336, 39005083331889816960, 1172419218038422659456, 37095226237402478348544

Offset: 0

N. J. A. Sloane, R. H. Hardin, Simon Plouffe

a(n+1) is the permanent of the n X n matrix M with M(i,i) = i+1, other entries 1. - Philippe Deléham, Nov 03 2005
Supernecklaces of type III (cycles of cycles). - Ricardo Bittencourt, May 05 2013
Unsigned coefficients for the raising / creation operator R for the Appell sequence of polynomials A238385: R = x + 1 - 2 D + 7 D^2/2! - 35 D^3/3! + ... . - Tom Copeland, May 09 2016

J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 125.
Jekuthiel Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 34
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 298

a(n)=|A039814(n, 1)| (first column of triangle). Cf. A000268, A000310, A000359, A000406, A001765.
Cf. A007840, A089064.
Cf. A238385.

series(ln(1/(1+ln(1-x))),x,17);
with (combstruct): M[ 1798 ] := [ A,{A=Cycle(Cycle(Z))},labeled ]:

With[{nn=20},CoefficientList[Series[Log[1/(1+Log[1-x])],{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Dec 15 2012 *)
Table[Sum[(-1)^(n-k) * (k-1)! * StirlingS1[n, k], {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 19 2024 *)

a(n)=if(n<0,0,n!*polcoeff(-log(1+log(1-x+x*O(x^n))),n))

Sum_{k=1..n} (k-1)!*|Stirling1(n, k)|. - Vladeta Jovovic, Sep 14 2003
a(n+1) = n! * Sum_{k=0..n} A007840(k)/k!. E.g., a(4) = 228 = 24*(1/1 + 1/1 + 3/2 + 14/6 + 88/24) = 24 + 24 + 36 + 56 + 88. - Philippe Deléham, Dec 10 2003
a(n) ~ (n-1)! * (exp(1)/(exp(1)-1))^n. - Vaclav Kotesovec, Jun 21 2013
a(0) = 0; a(n) = (n-1)! + Sum_{k=1..n-1} binomial(n-1,k) * (k-1)! * a(n-k). - Ilya Gutkovskiy, Jul 18 2020

Thanks to Paul Zimmermann for comments.

A048594 Triangle T(n,k) = k! * Stirling1(n,k), 1<=k<=n.

Original entry on oeis.org

1, -1, 2, 2, -6, 6, -6, 22, -36, 24, 24, -100, 210, -240, 120, -120, 548, -1350, 2040, -1800, 720, 720, -3528, 9744, -17640, 21000, -15120, 5040, -5040, 26136, -78792, 162456, -235200, 231840, -141120, 40320, 40320, -219168, 708744, -1614816, 2693880, -3265920, 2751840, -1451520, 362880

Offset: 1

Oleg Marichev (oleg(AT)wolfram.com)

Row sums (unsigned) give A007840(n), n>=1; (signed): A006252(n), n>=1.

Apart from signs, coefficients in expansion of n-th derivative of 1/log(x).

			Triangle begins
   1;
  -1,    2;
   2,   -6,   6;
  -6,   22, -36,   24;
  24, -100, 210, -240, 120; ...
The 2nd derivative of 1/log(x) is -2/x^3*log(x)^2 - 6/x^3*log(x)^3 - 6/x^3*log(x)^4.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
Eric Weisstein's World of Mathematics, Stirling Number of the First Kind
Wikipedia, Stirling numbers and exponential generating functions

Cf. A008275, A019538, A075181.
Cf. A133942 (left edge), A000142 (right edge), A006252 (row sums), A238685 (central terms).
Row sums: A007840 (unsigned), A006252 (signed).

a048594 n k = a048594_tabl !! (n-1) !! (k-1)
a048594_row n = a048594_tabl !! (n-1)
a048594_tabl = map snd $ iterate f (1, [1]) where
   f (i, xs) = (i + 1, zipWith (-) (zipWith (*) [1..] ([0] ++ xs))
                                   (map (* i) (xs ++ [0])))
-- Reinhard Zumkeller, Mar 02 2014

/* As triangle: */ [[Factorial(k)*StirlingFirst(n,k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Dec 15 2015

with(combinat): A048594 := (n,k)->k!*stirling1(n,k);

Flatten[Table[k!*StirlingS1[n,k], {n,10}, {k,n}]] (* Harvey P. Dale, Aug 28 2011 *)
Join @@ CoefficientRules[ -Table[ D[ 1/Log[z], {z, n}], {n, 9}] /. Log[z] -> -Log[z], {1/z, 1/Log[z]}, "NegativeLexicographic"][[All, All, 2]] (* Oleg Marichev (oleg(AT)wolfram.com) and Maxim Rytin (m.r(AT)inbox.ru); submitted by Robert G. Wilson v, Aug 29 2011 *)

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

def A048594(n,k): return (-1)^(n-k)*factorial(k)*stirling_number1(n,k)
flatten([[A048594(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Oct 24 2023

T(n, k) = k*T(n-1, k-1) - (n-1)*T(n-1, k) if n>=k>=1, T(n, 0) = 0 and T(1, 1)=1, else 0.
E.g.f. k-th column: log(1+x)^k, k>=1.
From Peter Bala, Nov 25 2011: (Start):
E.g.f.: 1/(1-t*log(1+x)) = 1 + t*x + (-t+2*t^2)*x^2/2! + ....
The row polynomials are given by D^n(1/(1-x*t)) evaluated at x = 0, where D is the operator exp(-x)*d/dx.
(End)

A088500 Expansion of e.g.f. 1/(1+2*log(1-x)).

Original entry on oeis.org

1, 2, 10, 76, 772, 9808, 149552, 2660544, 54093696, 1237306560, 31446049728, 879119219328, 26811313164672, 885830291432448, 31518653868782592, 1201567079771092992, 48860409899753588736, 2111033523652100407296

Offset: 0

Vladeta Jovovic, Nov 12 2003

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A007840, A088501.

CoefficientList[Series[1/(1+2*Log[1-x]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, May 03 2015 *)

my(x='x+O('x^30)); Vec(serlaplace(1/(1+2*log(1-x)))) \\ Michel Marcus, Apr 26 2021

a(n) = Sum_{k=0..n} |Stirling1(n, k)|*k!*2^k.
a(n) ~ n! * exp(n/2) / (2 * (exp(1/2)-1)^(n+1)). - Vaclav Kotesovec, May 03 2015
a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k). - Ilya Gutkovskiy, Apr 26 2021

A039814 Matrix square of Stirling-1 triangle A008275.

Original entry on oeis.org

1, -2, 1, 7, -6, 1, -35, 40, -12, 1, 228, -315, 130, -20, 1, -1834, 2908, -1485, 320, -30, 1, 17582, -30989, 18508, -5005, 665, -42, 1, -195866, 375611, -253400, 81088, -13650, 1232, -56, 1, 2487832, -5112570, 3805723, -1389612, 279048, -32130, 2100, -72, 1

Offset: 1

Christian G. Bower, Feb 15 1999

Exponential Riordan array [1/((1 + x)*(1 + log(1 + x))), log(1 + log(1 + x))]. The row sums of the unsigned array give A007840 (apart from the initial term). - Peter Bala, Jul 22 2014

Also the Bell transform of (-1)^n*A003713(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

			Triangle begins:
      1;
     -2,    1;
      7,   -6,     1;
    -35,   40,   -12,   1;
    228, -315,   130, -20,   1;
  -1834, 2908, -1485, 320, -30, 1;
...

Seiichi Manyama, Rows n = 1..140, flattened (Rows n = 1..60 from Vincenzo Librandi)

Column k=1..3 give (-1)^(n-1) * A003713(n), (-1)^n * A341587(n), (-1)^(n-1) * A341588(n).
Cf. A007840.
Cf. A039810, A039815, A039816, A039817.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (-1)^n*add(k!*abs(Stirling1(n+1,k+1)), k=0..n), 10); # Peter Luschny, Jan 28 2016

max = 9; t = Table[StirlingS1[n, k], {n, 1, max}, {k, 1, max}]; t2 = t.t; Table[t2[[n, k]], {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 01 2013 *)
rows = 9;
t = Table[(-1)^n*Sum[k!*Abs[StirlingS1[n+1, k+1]], {k,0,n}], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

T(n, k) = sum(j=0, n, stirling(n, j, 1)*stirling(j, k, 1)); \\ Seiichi Manyama, Feb 13 2022

E.g.f. of k-th column: ((log(1+log(1+x)))^k)/k!.
E.g.f.: 1/(1 + t)*( 1 + log(1 + t) )^(x-1) = 1 + (-2 + x)*t + (7 - 6*x + x^2)*t^2/2! + .... - Peter Bala, Jul 22 2014
T(n,k) = Sum_{j=0..n} Stirling1(n,j) * Stirling1(j,k). - Seiichi Manyama, Feb 13 2022

A052820 Expansion of e.g.f. 1/(1 - x + log(1 - x)).

Original entry on oeis.org

1, 2, 9, 62, 572, 6604, 91526, 1480044, 27353448, 568731648, 13138994112, 333895239072, 9256507508112, 278000959058016, 8991458660924112, 311585506208924064, 11517363473843526912, 452332548042633835776

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Previous name was: A simple grammar.

a(n) is the number of ways to seat n people at circular tables, then linearly order the tables, then designate some (possibly all or none) of the tables at which only one person is seated. a(2) = 9 because we have: (1)(2), (1')(2), (1)(2'), (1')(2'), (2)(1), (2')(1), (2)(1'), (2')(1'), (1,2). Cf. A007840. - Geoffrey Critzer, Nov 05 2013

Seiichi Manyama, Table of n, a(n) for n = 0..395
W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 Mar 2013.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 785
Makhin Thitsa and W. Steven Gray, On the Radius of Convergence of Cascaded Analytic Nonlinear Systems, 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), Orlando, FL, USA, December 12-15, 2011, pp. 3830-3835.
M. Thitsa and W. S. Gray, On the radius of convergence of cascaded analytic nonlinear systems: The SISO case, System Theory (SSST), 2011 IEEE 43rd Southeastern Symposium on, 14-16 March 2011, pp. 30-36.
Makhin Thitsa and W. Steven Gray, On the Radius of Convergence of Interconnected Analytic Nonlinear Input-Output Systems, SIAM Journal on Control and Optimization, Vol. 50, No. 5, 2012, pp. 2786-2813. - From _N. J. A. Sloane_, Dec 26 2012

Cf. A030178.
Cf. A367851, A367852, A367853.
Cf. A367845, A367846, A367847.

spec := [S,{C=Cycle(Z),B=Union(C,Z),S=Sequence(B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

CoefficientList[Series[1/(1-x+Log[1-x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2013 *)

E.g.f.: -1/(-1+x+log(-1/(-1+x))).
a(n) ~ n! * (1/(1-LambertW(1)))^n/(1/LambertW(1)-LambertW(1)). - Vaclav Kotesovec, Oct 01 2013
a(0) = 1; a(n) = n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k). - Ilya Gutkovskiy, Apr 26 2021

New name using e.g.f., Vaclav Kotesovec, Oct 01 2013

A052801 A simple grammar: labeled pairs of sequences of cycles.

Original entry on oeis.org

1, 2, 8, 46, 342, 3108, 33324, 411360, 5741856, 89379120, 1534623936, 28804923024, 586686138384, 12885385945248, 303537419684064, 7633673997722496, 204125888803996800, 5782960189212871680

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Robert Israel, Table of n, a(n) for n = 0..417
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 759.

Cf. A215916.

Cf. A007840, A354122, A354123.

spec := [S,{C=Cycle(Z),B=Sequence(C),S=Prod(B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

CoefficientList[Series[1/(1+Log[1-x])^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)

makelist(sum((-1)^(n-k)*stirling1(n, k)*(k+1)!, k, 0, n), n, 0, 17); /* Bruno Berselli, May 25 2011 */

E.g.f.: 1/(-1+log(-1/(-1+x)))^2.
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling1(n, k)*(k+1)!. - Vladeta Jovovic, Sep 21 2003
a(n) = D^n(1/(1-x)^2) evaluated at x = 0, where D is the operator exp(x)*d/dx. Cf. A052811. - Peter Bala, Nov 25 2011
a(n) ~ n! * n*exp(n)/(exp(1)-1)^(n+2). - Vaclav Kotesovec, Sep 30 2013
From Anton Zakharov, Aug 07 2016: (Start)
a(n) = A007840(n) + A215916(n).
a(n) = Sum_{k=2..n+1} k!*s(n,k) where s(n,k) is the unsigned Stirling number of the first kind, (A132393). (End)
a(0) = 1; a(n) = Sum_{k=1..n} (k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023

A225479 Triangle read by rows, the ordered Stirling cycle numbers, T(n, k) = k!* s(n, k); n >= 0 k >= 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 6, 6, 0, 6, 22, 36, 24, 0, 24, 100, 210, 240, 120, 0, 120, 548, 1350, 2040, 1800, 720, 0, 720, 3528, 9744, 17640, 21000, 15120, 5040, 0, 5040, 26136, 78792, 162456, 235200, 231840, 141120, 40320, 0, 40320, 219168, 708744, 1614816

Offset: 0

Peter Luschny, May 20 2013

The Digital Library of Mathematical Functions defines the Stirling cycle numbers as (-1)^(n-k) times the Stirling numbers of the first kind.

			[n\k][0,   1,   2,    3,    4,    5,   6]
[0]   1,
[1]   0,   1,
[2]   0,   1,   2,
[3]   0,   2,   6,    6,
[4]   0,   6,  22,   36,   24,
[5]   0,  24, 100,  210,  240,  120,
[6]   0, 120, 548, 1350, 2040, 1800, 720.
...
T(4,2) = 22: The table below shows the compositions of 4 into two parts.
n = 4    Composition       Weight     4!*Weight
            3 + 1            1/3         8
            1 + 3            1/3         8
            2 + 2          1/2*1/2       6
                                        = =
                                  total 22

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, table 245.

Vincenzo Librandi, Rows n = 0..50, flattened
Digital Library of Mathematical Functions, Set Partitions: Stirling Numbers
S. Eger, Restricted Weighted Integer Compositions and Extended Binomial Coefficients J. Integer. Seq., Vol. 16 (2013), Article 13.1.3

Cf. A048594 (signed version without the first column), A132393.

Columns k=0..6 are A000007, A104150, A052517, A052748, A052753, A052767, A052779.

A225479 := proc(n, k) option remember;
if k > n or  k < 0 then return(0) fi;
if n = 0 and k = 0 then return(1) fi;
k*A225479(n-1, k-1) + (n-1)*A225479(n-1, k) end;
for n from 0 to 9 do seq(A225479(n, k), k = 0..n) od;

t[n_, k_] := k!*StirlingS1[n, k] // Abs; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 02 2013 *)

T(n,k)={k!*abs(stirling(n,k,1))} \\ Andrew Howroyd, Jul 27 2020

def A225479(n, k): return factorial(k)*stirling_number1(n, k)
for n in (0..6): [A225479(n,k) for k in (0..n)]

For a recursion see the Maple program.
T(n, 0) = A000007; T(n, 1) = A000142; T(n, 2) = A052517.
T(n, 3) = A052748; T(n, n) = A000142; T(n, n-1) = A001286.
row sums = A007840; alternating row sums = A006252.
From Peter Bala, Sep 20 2013: (Start)
E.g.f.: 1/(1 + x*log(1 - t)) = 1 + x*t + (x + 2*x^2)*t^2/2! + (2*x + 6*x^2 + 6*x^3)*t^3/3! + ....
T(n,k) = n!*( the sum of the total weight of the compositions of n into k parts where each part i has weight 1/i ) (see Eger, Theorem 1). An example is given below.  (End)
T(n,k) = A132393(n,k) * A000142(k). - Philippe Deléham, Jun 24 2015

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.

A320096 a(n) = Sum_{k=1..n} (-1)^(n-k) * Stirling1(n,k) * k! * k^n, with a(0)=1.

Original entry on oeis.org

1, 1, 9, 212, 9418, 675014, 71092502, 10334690232, 1982433606264, 485065343565072, 147433546709109408, 54493722609862927632, 24069397682825072219040, 12520250948941157091235344, 7575515622713954399390221008, 5275250174853125498317783254528

Offset: 0

Vaclav Kotesovec, Oct 05 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..229

Cf. A007840, A220181, A320083.

Flatten[{1, Table[Sum[(-1)^(n-k)*StirlingS1[n, k]*k!*k^n, {k, 1, n}], {n, 1, 20}]}]
nmax = 20; CoefficientList[Series[1 + Sum[(-Log[1 - k*x])^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 04 2022 *)

a(n) = sum(k=0, n, (-1)^(n-k)*k!*k^n*stirling(n, k, 1)); \\ Seiichi Manyama, Feb 02 2022

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-log(1-k*x))^k))) \\ Seiichi Manyama, Feb 02 2022

a(n) ~ c * d^n * n^(2*n + 1/2), where
w = -LambertW(-1, -exp(-r)*r) = 1.1628296650659469964248518258036278907318113...
r = 0.8531304407911771560472963194514988627832723535823134189532... is the real root of the equation w = r + exp(-1/r)
d = exp(-1)*r*w*(w-r)^(r-1) = 0.433513333588184444899487502412976956849408575992...
c = 1.959633090979666812031505093625147349925787002426082...
E.g.f.: Sum_{k>=0} (-log(1 - k*x))^k. - Seiichi Manyama, Feb 02 2022

cp's OEIS Frontend

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