A008296 -id:A008296 - cp's OEIS frontend

A045406 A diagonal of A008296.

1, 3, -1, 0, 4, -28, 188, -1368, 11016, -98208, 964512, -10370880, 121337280, -1535880960, 20924455680, -305396421120, 4755302899200, -78700195123200, 1379748896870400, -25546854999859200, 498194992408780800, -10207190048993280000, 219216795045212160000

Offset: 2

N. J. A. Sloane, Jan 26 2001

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 139, b(n,2).

Robert Israel, Table of n, a(n) for n = 2..414

Cf. A081048.

with(combinat): for n from 2 to 40 do for k from 2 to 2 do printf(`%d,`,sum(binomial(l,k)*k^(l-k)*stirling1(n,l), l=k..n)) od: od:
# Alternative:
A081048:= gfun:-rectoproc({a(0)=0,a(1)=1,a(n)=(1-2*n)*a(n-1) -(n-1)^2*a(n-2)},a(n),remember):
1, seq(2*A081048(n-3)-3*(-1)^(n)*(n-3)!,n=3..50); # Robert Israel, Jun 29 2015

With[{nn=30},CoefficientList[Series[((1+x)Log[1+x])^2/2,{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jun 04 2019 *)

a(n) = A008296(n,2).
E.g.f.: ((1+x)*log(1+x))^2/2. - Vladeta Jovovic, Feb 20 2003
a(n) = sum(i=1, n-1, i^2*Stirling1(n-1, i)). - Benoit Cloitre, Oct 23 2004
If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then a(n) = f(n,2,-2), for n>=2. - Milan Janjic, Dec 21 2008
a(n) = (-1)^(n)*(2*H(n-3)-3)*(n-3)! for n >= 3, where H(n) = Sum(j=1..n, 1/j) is the n-th harmonic number. - Gary Detlefs, Feb 13 2010
a(n) = 2*A081048(n-3)-3*(-1)^(n)*(n-3)! for n >= 3. - Robert Israel, Jun 28 2015
Sum_{k=1..n} a(k+1) * Stirling2(n,k) = n^2. - Vaclav Kotesovec, Sep 03 2018
Conjecture: D-finite with recurrence a(n) +(2*n-7)*a(n-1) +(n-4)^2*a(n-2)=0. - R. J. Mathar, Sep 15 2021

More terms from James Sellers, Jan 26 2001

Gary Detlefs comment changed to a formula by Robert Israel, Jun 28 2015

A059302 A diagonal of A008296.

Original entry on oeis.org

-1, -1, 5, 25, 70, 154, 294, 510, 825, 1265, 1859, 2639, 3640, 4900, 6460, 8364, 10659, 13395, 16625, 20405, 24794, 29854, 35650, 42250, 49725, 58149, 67599, 78155, 89900, 102920, 117304, 133144, 150535, 169575, 190365, 213009, 237614, 264290, 293150, 324310, 357889, 394009, 432795, 474375

Offset: 2

N. J. A. Sloane, Jan 26 2001

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 139, b(n,n-2).

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. similar sequences listed in A241765.

with(combinat): for n from 3 to 100 do for k from n-2 to n-2 do printf(`%d,`,sum(binomial(l,k)*k^(l-k)*stirling1(n,l), l=k..n)) od: od:

f[n_] := 3*n - 1; s1 = s2 = s3 = 0; lst = {}; Do[a = f[n]; s1 += a;
s2 += s1; s3 += s2; AppendTo[lst, s3], {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jun 27 2009 *)
Drop[CoefficientList[Series[-x^2 (1 - 4*x)/(1 - x)^5, {x, 0, 50}], x], 2] (* Vincenzo Librandi, Mar 22 2012 *)
Rest[Table[(n - 1) n (n + 1)(3 n - 10)/24, {n, 50}]] (* or *) LinearRecurrence[ {5, -10, 10, -5, 1},{-1, -1, 5, 25, 70}, 50] (* Harvey P. Dale, Jun 05 2012 *)

x='x+O('x^99); Vec(x^2*(-1+4*x)/(1-x)^5) \\ Altug Alkan, Sep 13 2017

a(n) = (n-1)n(n+1)(3n-10)/24.
If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-a-j), then a(n-1) = f(n,n-2,-1), for n >= 3. - Milan Janjic, Dec 20 2008
G.f.: -x^2*(1-4*x)/(1-x)^5. - Colin Barker, Mar 21 2012
a(2)=-1, a(3)=-1, a(4)=5, a(5)=25, a(6)=70, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Jun 05 2012
a(n) = Sum_{k=1..n} Sum_{i=1..k} (n-i)*(n-k-1). - Wesley Ivan Hurt, Sep 12 2017

More terms from James Sellers, Jan 26 2001

A298511 Central Lehmer-Comtet numbers of the first kind: a(n) = A008296(2n,n).

Original entry on oeis.org

1, 1, -1, -15, 49, 1365, -7645, -311311, 2475473, 132652377, -1367593305, -90881245455, 1151541572401, 91341008892445, -1373222414339685, -126594821384553375, 2202549127844351265, 231390624855674406705, -4573116447815658471025, -539278542630309415030735

Offset: 0

Alois P. Heinz, Jan 20 2018

sign

Alois P. Heinz, Table of n, a(n) for n = 0..386

Cf. A008275, A008296, A048994.

b:= proc(n, k) option remember; `if`(n=k, 1, `if`(k=0, 0,
      (n-1)*b(n-2, k-1)+b(n-1, k-1)+(k-n+1)*b(n-1, k)))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..25);

b[n_, k_] := b[n, k] = If[n == k, 1, If[k == 0, 0,
   (n-1) b[n-2, k-1] + b[n-1, k-1] + (k-n+1) b[n-1, k]]];
a[n_] := b[2n, n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Dec 01 2023, from Maple code *)

a(n) = (2*n)!/n! * [x^(2*n)] ((1+x)*log(1+x))^n.

a(n) = Sum_{j=0..n} binomial(n+j,n) * n^j * Stirling1(2*n,n+j).

A345651 Fourth column of A008296.

Original entry on oeis.org

1, 10, 25, -35, 49, 0, -820, 9020, -87164, 859144, -8965320, 100136400, -1199838576, 15406135488, -211479420096, 3094582896000, -48129022468224, 793274283938304, -13818265424460288, 253731538514893824, -4899371564756837376, 99261476593521868800

Offset: 4

Luca Onnis, Aug 26 2021

sign

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 139, b(n,4).

Cf. A008296, A045406, A081048.

Cf. A001008, A002805, A007406, A007407.

b:= proc(n, k) option remember; `if`(n=k, 1, `if`(k=0, 0,
      (n-1)*b(n-2, k-1)+b(n-1, k-1)+(k-n+1)*b(n-1, k)))
    end:
a:= n-> b(n, 4):
seq(a(n), n=4..28);  # Alois P. Heinz, Aug 26 2021
# alternative
seq(A008296(n,4),n=4..70) ; # R. J. Mathar, Sep 15 2021

a[1, 1] = a[2, 1] = 1; a[n_, 1] = (-1)^n (n - 2)!;
a[n_, n_] = 1;
a[n_, k_] := a[n, k] = (n - 1) a[n - 2, k - 1] +
    a[n - 1, k - 1] + (k - n + 1) a[n - 1, k];
Flatten[Table[N[a[n + 4, 4], 10], {n, 1, 400}]]

a(n) = sum(m=4, n, binomial(m, 4)*4^(m-4)*stirling(n, m, 1)); \\ Michel Marcus, Sep 14 2021

a(n) = A008296(n,4).
a(n) = (-1)^n*(4*H(n-5,1)^3 + 8*H(n-5,3) - 12*H(n-5,2)*H(n-5,1) - 25*H(n-5,1)^2 + 25*H(n-5,2) + 35*H(n-5,1) - 10)*(n-5)! for n >= 5 where H(n,1) = Sum_{j=1..n} 1/j is the n-th harmonic number, H(n,2) = Sum_{j=1..n} 1/j^2 and H(n,3) = Sum_{j=1..n} 1/j^3.
a(n) = Sum_{m=4..n} binomial(m,4) * 4^(m-4) * Stirling1(n,m). - Alois P. Heinz, Aug 26 2021
Conjecture: D-finite with recurrence a(n) +2*(2*n-13)*a(n-1) +(6*n^2-84*n+295)*a(n-2) +(2*n-15)*(2*n^2-30*n+113)*a(n-3) +(n-8)^4*a(n-4)=0. - R. J. Mathar, Sep 15 2021

A347276 Third column of A008296.

Original entry on oeis.org

1, 6, 5, -15, 49, -196, 944, -5340, 34716, -254760, 2078856, -18620784, 180973584, -1887504768, 20887922304, -242111586816, 2889841121280, -34586897978880, 393722260047360, -3659128846433280, 5687630494110720, 1137542166526464000, -49644151627682304000

Offset: 3

Luca Onnis, Aug 25 2021

sign

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 139, b(n,3).

Cf. A008296, A045406, A081048.
Cf. A001008, A002805, A007406, A007407.
Cf. A048994, A008275.

b:= proc(n, k) option remember; `if`(n=k, 1, `if`(k=0, 0,
      (n-1)*b(n-2, k-1)+b(n-1, k-1)+(k-n+1)*b(n-1, k)))
    end:
a:= n-> b(n, 3):
seq(a(n), n=3..30);  # Alois P. Heinz, Aug 25 2021

a[1, 1] = a[2, 1] = 1; a[n_, 1] = (-1)^n (n - 2)!;
a[n_, n_] = 1;
a[n_, k_] :=  a[n, k] = (n - 1) a[n - 2, k - 1] +
    a[n - 1, k - 1] + (k - n + 1) a[n - 1, k];
Flatten[Table[a[n + 3, 3], {n, 0, 30}]]

a(n) = sum(m=3, n, binomial(m, 3)*3^(m-3)*stirling(n, m, 1)); \\ Michel Marcus, Sep 14 2021

a(n) = A008296(n,3).
a(n) = (-1)^n*(3*H(n-4,1)^2 - 3*H(n-4,2) - 11*H(n-4,1) + 6)*(n-4)! for n >= 4, where H(n,1) = Sum_{j=1..n} 1/j = A001008(n)/A002805(n) is the n-th harmonic number and H(n,2) = Sum_{j=1..n} 1/j^2 = A007406(n)/A007407(n).
a(n) = Sum_{m=3..n} binomial(m,3) * 3^(m-3) * Stirling1(n,m). - Alois P. Heinz, Aug 26 2021

A039620 Erroneous version of A008296.

Original entry on oeis.org

1, 1, 1, -1, 3, 1, 2, -1, 6, 1, -6, 0, 4, 5, 10, 1, 24, 4, -15, 25, 15, 1, -120, -28, 49, -35

Offset: 1

dead

A008297 Triangle of Lah numbers.

Original entry on oeis.org

-1, 2, 1, -6, -6, -1, 24, 36, 12, 1, -120, -240, -120, -20, -1, 720, 1800, 1200, 300, 30, 1, -5040, -15120, -12600, -4200, -630, -42, -1, 40320, 141120, 141120, 58800, 11760, 1176, 56, 1, -362880, -1451520, -1693440, -846720, -211680, -28224, -2016, -72, -1, 3628800, 16329600, 21772800, 12700800

Offset: 1

N. J. A. Sloane

|a(n,k)| = number of partitions of {1..n} into k lists, where a list means an ordered subset.
Let N be a Poisson random variable with parameter (mean) lambda, and Y_1,Y_2,... independent exponential(theta) variables, independent of N, so that their density is given by (1/theta)*exp(-x/theta), x > 0. Set S=Sum_{i=1..N} Y_i. Then E(S^n), i.e., the n-th moment of S, is given by (theta^n) * L_n(lambda), n >= 0, where L_n(y) is the Lah polynomial Sum_{k=0..n} |a(n,k)| * y^k. - Shai Covo (green355(AT)netvision.net.il), Feb 09 2010
For y = lambda > 0, formula 2) for the Lah polynomial L_n(y) dated Feb 02 2010 can be restated as follows: L_n(lambda) is the n-th ascending factorial moment of the Poisson distribution with parameter (mean) lambda. - Shai Covo (green355(AT)netvision.net.il), Feb 10 2010
See A111596 for an expression of the row polynomials in terms of an umbral composition of the Bell polynomials and relation to an inverse Mellin transform and a generalized Dobinski formula. - Tom Copeland, Nov 21 2011
Also the Bell transform of the sequence (-1)^(n+1)*(n+1)! without column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016
Named after the Slovenian mathematician and actuary Ivo Lah (1896-1979). - Amiram Eldar, Jun 13 2021

			|a(2,1)| = 2: (12), (21); |a(2,2)| = 1: (1)(2). |a(4,1)| = 24: (1234) (24 ways); |a(4,2)| = 36: (123)(4) (6*4 ways), (12)(34) (3*4 ways); |a(4,3)| = 12: (12)(3)(4) (6*2 ways); |a(4,4)| = 1: (1)(2)(3)(4) (1 way).
Triangle:
    -1;
     2,    1;
    -6,   -6,   -1;
    24,   36,   12,   1;
  -120, -240, -120, -20, -1; ...

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
Shai Covo, The moments of a compound Poisson process with exponential or centered normal jumps, J. Probab. Stat. Sci., Vol. 7, No. 1 (2009), pp. 91-100.
Theodore S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176; the sequence called {!}^{n+}. For a link to this paper see A000262.
John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.
S. Gill Williamson, Combinatorics for Computer Science, Computer Science Press, 1985; see p. 176.

T. D. Noe, Rows n=1..100 of triangle, flattened
J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013.
P. Blasiak, K. A. Penson, and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers, arXiv:quant-ph/0212072, 2002.
P. Blasiak, K. A. Penson, and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015.
Tom Copeland, Lagrange a la Lah, 2011.
Siad Daboul, Jan Mangaldan, Michael Z. Spivey, and Peter J. Taylor, The Lah Numbers and the n-th Derivative of exp(1/x), Math. Mag., Vol. 86, No. 1 (2013), pp. 39-47.
Askar Dzhumadil'daev and Damir Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1 [math.CO], 2014-2015. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
Askar Dzhumadil’daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, Vol. 22, No. 4 (2015), #P4.10.
B. S. El-Desouky, Nenad P.Cakić, and Toufik Mansour, Modified approach to generalized Stirling numbers via differential operators, Appl. Math. Lett., Vol. 23, No. 1 (2010), pp. 115-120.
Sen-Peng Eu, Tung-Shan Fu, Yu-Chang Liang, and Tsai-Lien Wong. On xD-Generalizations of Stirling Numbers and Lah Numbers via Graphs and Rooks. arXiv:1701.00600 [math.CO], 2017.
Milan Janjic, Some classes of numbers and derivatives, JIS, Vol. 12 (2009), pp. 09.8.3
Dmitry Karp and Elena Prilepkina, Generalized Stieltjes transforms: basic aspects, arXiv preprint arXiv:1111.4271 [math.CA], 2011.
Dmitry Karp and Elena Prilepkina, Generalized Stieltjes functions and their exact order, Journal of Classical Analysis, Vol. 1, No. 1 (2012), pp. 53-74. - _N. J. A. Sloane_, Dec 25 2012.
Udita N. Katugampola, A new Fractional Derivative and its Mellin Transform, arXiv preprint arXiv:1106.0965 [math.CA], 2011.
Udita N. Katugampola, Mellin Transforms of the Generalized Fractional Integrals and Derivatives, arXiv preprint arXiv:1112.6031 [math.CA], 2011.
Donald E. Knuth, Convolution polynomials, Mathematica J. 2.1 (1992), no. 4, 67-78; arXiv:math/9207221 [math.CA], 1992.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seq., Vol. 3 (2000), Article 00.2.4.
Toufik Mansour and Matthias Schork, Generalized Bell numbers and algebraic differential equations, Pure Math. Appl.(PU. MA), Vol. 23, No. 2 (2012), pp. 131-142.
Toufik Mansour, Matthias Schork, and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), Article 12.8.3.
Toufik Mansour and Mark Shattuck, A polynomial generalization of some associated sequences related to set partitions, Periodica Mathematica Hungarica, Vol. 75, No. 2 (December 2017), pp. 398-412.
Toufik Mansour, Augustine Munagi, and Mark Shattuck, Set partitions with colored singleton blocks, Discrete Mathematics Letters, 13. 100, (2024). See p. 100.
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics, Vol. 13, No. 2 (2019), pp. 495-517.
Mathias Pétréolle and Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019.
Jose L. Ramirez and Mark Shattuck, A (p, q)-Analogue of the r-Whitney-Lah Numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.6.
Mark Shattuck, Identities for Generalized Whitney and Stirling Numbers, Journal of Integer Sequences, Vol. 20 (2017), Article 17.10.4.
Mark Shattuck, Some formulas for the restricted r-Lah numbers, Annales Mathematicae et Informaticae, Vol. 49 (2018), Eszterházy Károly University Institute of Mathematics and Informatics, pp. 123-140.
Mark Shattuck, Combinatorial Proofs of Some Stirling Number Convolution Formulas, J. Int. Seq., Vol. 25 (2022), Article 22.2.2.
Michael Z. Spivey, On Solutions to a General Combinatorial Recurrence, J. Int. Seq., Vol. 14 (2011) Article 11.9.7.
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.
Bao-Xuan Zhu, Total positivity from a generalized cycle index polynomial, arXiv:2006.14485 [math.CO], 2020.

Same as A066667 and A105278 except for signs. Other variants: A111596 (differently signed triangle and (0,0)-based), A271703 (unsigned and (0,0)-based), A089231.
A293125 (row sums) and A000262 (row sums of unsigned triangle).
Columns 1-6 (unsigned): A000142, A001286, A001754, A001755, A001777, A001778.
A002868 gives maximal element (in magnitude) in each row.
A248045 (central terms, negated). A130561 is a natural refinement.
Cf. A007318, A048786, A001263, A008275, A008277, A048993, A094638, A130534.
Cf. A008296, A360177.

a008297 n k = a008297_tabl !! (n-1) !! (k-1)
a008297_row n = a008297_tabl !! (n-1)
a008297_tabl = [-1] : f [-1] 2 where
   f xs i = ys : f ys (i + 1) where
     ys = map negate $
          zipWith (+) ([0] ++ xs) (zipWith (*) [i, i + 1 ..] (xs ++ [0]))
-- Reinhard Zumkeller, Sep 30 2014

A008297 := (n,m) -> (-1)^n*n!*binomial(n-1,m-1)/m!;

a[n_, m_] := (-1)^n*n!*Binomial[n-1, m-1]/m!; Table[a[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Jean-François Alcover, Dec 12 2012, after Maple *)
T[n_, n_] := (-1)^n; T[n_, k_]/;0Oliver Seipel, Dec 06 2024 *)

T(n, m) = (-1)^n*n!*binomial(n-1, m-1)/m!
for(n=1,9, for(m=1,n, print1(T(n,m)", "))) \\ Charles R Greathouse IV, Mar 09 2016

use bigint; use ntheory ":all"; my @L; for my $n (1..9) { push @L, map { stirling($n,$,3)*(-1)**$n } 1..$n; } say join(", ",@L); # _Dana Jacobsen, Mar 16 2017

def A008297_triangle(dim): # computes unsigned T(n, k).
    M = matrix(ZZ,dim,dim)
    for n in (0..dim-1): M[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n,k] = M[n-1,k-1]+(2+2*k)*M[n-1,k]+((k+1)*(k+2))*M[n-1,k+1]
    return M
A008297_triangle(9) # Peter Luschny, Sep 19 2012

a(n, m) = (-1)^n*n!*A007318(n-1, m-1)/m!, n >= m >= 1.
a(n+1, m) = (n+m)*a(n, m)+a(n, m-1), a(n, 0) := 0; a(n, m) := 0, n < m; a(1, 1)=1.
a(n, m) = ((-1)^(n-m+1))*L(1, n-1, m-1) where L(1, n, m) is the triangle of coefficients of the generalized Laguerre polynomials n!*L(n, a=1, x). These polynomials appear in the radial l=0 eigen-functions for discrete energy levels of the H-atom.
|a(n, m)| = Sum_{k=m..n} |A008275(n, k)|*A008277(k, m), where A008275 = Stirling numbers of first kind, A008277 = Stirling numbers of second kind. - Wolfdieter Lang
If L_n(y) = Sum_{k=0..n} |a(n, k)|*y^k (a Lah polynomial) then the e.g.f. for L_n(y) is exp(x*y/(1-x)). - Vladeta Jovovic, Jan 06 2001
E.g.f. for the k-th column (unsigned): x^k/(1-x)^k/k!. - Vladeta Jovovic, Dec 03 2002
a(n, k) = (n-k+1)!*N(n, k) where N(n, k) is the Narayana triangle A001263. - Philippe Deléham, Jul 20 2003
From Shai Covo (green355(AT)netvision.net.il), Feb 02 2010: (Start)
We have the following expressions for the Lah polynomial L_n(y) = Sum_{k=0..n} |a(n, k)|*y^k -- exact generalizations of results in A000262 for A000262(n) = L_n(1):
1) L_n(y) = y*exp(-y)*n!*M(n+1,2,y), n >= 1, where M (=1F1) is the confluent hypergeometric function of the first kind;
2) L_n(y) = exp(-y)* Sum_{m>=0} y^m*[m]^n/m!, n>=0, where [m]^n = m*(m+1)*...*(m+n-1) is the rising factorial;
3) L_n(y) = (2n-2+y)L_{n-1}(y)-(n-1)(n-2)L_{n-2}(y), n>=2;
4) L_n(y) = y*(n-1)!*Sum_{k=1..n} (L_{n-k}(y) k!)/((n-k)! (k-1)!), n>=1. (End)
The row polynomials are given by D^n(exp(-x*t)) evaluated at x = 0, where D is the operator (1-x)^2*d/dx. Cf. A008277 and A035342. - Peter Bala, Nov 25 2011
n!C(-xD,n) = Lah(n,:xD:) where C(m,n) is the binomial coefficient, xD= x d/dx, (:xD:)^k = x^k D^k, and Lah(n,x) are the row polynomials of this entry. E.g., 2!C(-xD,2)= 2 xD + x^2 D^2. - Tom Copeland, Nov 03 2012
From Tom Copeland, Sep 25 2016: (Start)
The Stirling polynomials of the second kind A048993 (A008277), i.e., the Bell-Touchard-exponential polynomials B_n[x], are umbral compositional inverses of the Stirling polynomials of the first kind signed A008275 (A130534), i.e., the falling factorials, (x)_n = n! binomial(x,n); that is, umbrally B_n[(x).] = x^n = (B.[x])_n.
An operational definition of the Bell polynomials is (xD_x)^n = B_n[:xD:], where, by definition, (:xD_x:)^n = x^n D_x^n, so (B.[:xD_x:])_n = (xD_x)_n = :xD_x:^n = x^n (D_x)^n.
Let y = 1/x, then D_x = -y^2 D_y; xD_x = -yD_y; and P_n(:yD_y:) = (-yD_y)_n = (-1)^n (1/y)^n (y^2 D_y)^n, the row polynomials of this entry in operational form, e.g., P_3(:yD_y:) = (-yD_y)_3 = (-yD_y) (yD_y-1) (yD_y-2) = (-1)^3 (1/y)^3 (y^2 D_y)^3 = -( 6 :yD_y: +  6 :yD_y:^2 + :yD_y:^3 ) = - ( 6 y D_y + 6 y^2 (D_y)^2 + y^3 (D_y)^3).
Therefore, P_n(y) = e^(-y) P_n(:yD_y:) e^y = e^(-y) (-1/y)^n (y^2 D_y)^n e^y = e^(-1/x) x^n (D_x)^n e^(1/x) = P_n(1/x) and P_n(x) =  e^(-1/x) x^n (D_x)^n e^(1/x) = e^(-1/x) (:x D_x:)^n e^(1/x). (Cf. also A094638.) (End)
T(n,k) = Sum_{j=k..n} (-1)^j*A008296(n,j)*A360177(j,k). - Mélika Tebni, Feb 02 2023

A005727 n-th derivative of x^x at x=1. Also called Lehmer-Comtet numbers.

Original entry on oeis.org

1, 1, 2, 3, 8, 10, 54, -42, 944, -5112, 47160, -419760, 4297512, -47607144, 575023344, -7500202920, 105180931200, -1578296510400, 25238664189504, -428528786243904, 7700297625889920, -146004847062359040, 2913398154375730560, -61031188196889482880

Offset: 0

N. J. A. Sloane

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 139, table at foot of page.
G. H. Hardy, A Course of Pure Mathematics, 10th ed., Cambridge University Press, 1960, p. 428.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..400 (first 101 terms from T. D. Noe)
Joerg Arndt, Matters Computational (The Fxtbook), section 36.5, "The function x^x"
H. W. Gould, A Set of Polynomials Associated with the Higher Derivatives of y=xxy=x^x, Rocky Mountain J. Math. 26(2) 1996.
R. K. Guy, Letter to N. J. A. Sloane, 1986
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
G. H. Hardy, A Course of Pure Mathematics, Cambridge, The University Press, 1908.
D. H. Lehmer, Numbers associated with Stirling Numbers and x^x, Rocky Mountain J. Math., 15(2) 1985, p. 461.
R. R. Patterson and G. Suri, The derivatives of x^x, date unknown. Preprint. [Annotated scanned copy]

Row sums of A008296. Column k=2 of A215703 and of A277537.

Cf. A005168, A176118, A293297, A203852.

A005727 := proc(n) option remember; `if`(n=0, 1, A005727(n-1)+add((-1)^(n-k)*(n-2-k)!*binomial(n-1, k)*A005727(k), k=0..n-2)) end:
seq(A005727(n), n=0..23); # Mélika Tebni, May 22 2022

NestList[ Factor[ D[ #1, x ] ]&, x^x, n ] /. (x->1)
Range[0, 22]! CoefficientList[ Series[(1 + x)^(1 + x), {x, 0, 22}], x] (* Robert G. Wilson v, Feb 03 2013 *)

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

For n>0, a(n) = Sum_{k=0..n} b(n, k), where b(n, k) is a Lehmer-Comtet number of the first kind (see A008296).
E.g.f.: (1+x)^(1+x). a(n) = Sum_{k=0..n} Stirling1(n, k)*A000248(k). - Vladeta Jovovic, Oct 02 2003
From  Mélika Tebni, May 22 2022: (Start)
a(0) = 1, a(n) = a(n-1)+Sum_{k=0..n-2} (-1)^(n-k)*(n-2-k)!*binomial(n-1, k)*a(k).
a(n) = Sum_{k=0..n} (-1)^(n-k)*A293297(k)*binomial(n, k).
a(n) = Sum_{k=0..n} (-1)^k*A203852(k)*binomial(n, k). (End)

A176118 The n-th derivative of 1/x^x, evaluated at x=1.

Original entry on oeis.org

1, -1, 0, 3, -8, 10, 6, -42, -160, 2952, -27720, 253440, -2553528, 28562664, -349272000, 4618376280, -65615072640, 996952226880, -16133983959744, 277093189849536, -5033937521116800, 96451913892983040, -1943937259314019200, 41112770486238380160

Offset: 0

Jacob Parr (jacobparr1(AT)gmail.com), Apr 09 2010

sign

			E.g.f.: A(x) = 1 - x + 3*x^3/3! - 8*x^4/4! + 10*x^5/5! + 6*x^6/6! - 42*x^7/7! - 160*x^8/8! + 2952*x^9/9! - 27720*x^10/10! + 253440*x^11/11! + ...
The e.g.f. as a power series with reduced fractional coefficients begins
A(x) = 1 - x + 1/2x^3 - 1/3x^4 + 1/12x^5 + 1/120x^6 - 1/120x^7 - 1/252x^8 + 41/5040x^9 - 11/1440x^10 + 2/315x^11 - 106397/19958400x^12 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A005727, A008296, A194786.

1, seq(simplify(subs(x = 1, diff(x^(-x), `$`(x, n)))), n = 1 .. 22); # Emeric Deutsch, Apr 14 2010
a:= n-> n! *coeftayl(x^(-x), x=1, n):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 18 2012

NestList[Factor[D[#1, x]] &, 1/x^x, 22] /. (x -> 1) (* Robert G. Wilson v, Feb 03 2013 *)

E.g.f.: 1 + x*(Q(0) - 1)/(x+1) where Q(k) = 1 - (1+x/(k+1))/(1 - x/(x + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Mar 05 2013
a(n) ~ (-1)^(n+1) * n! / n^2. - Vaclav Kotesovec, Sep 03 2014
E.g.f.: 1/(x+1)^(x+1). - Alois P. Heinz, Sep 27 2016
a(n) = Sum_{k=0..n} (-1)^k * A008296(n,k). - Alois P. Heinz, Aug 25 2021
E.g.f.: Sum_{n>=0} (-1)^n * x^n/n! * Product_{k=1..n} (k + x). - Paul D. Hanna, Nov 13 2023

Definition edited by Emeric Deutsch, Apr 14 2010

More terms from Emeric Deutsch and R. J. Mathar, Apr 14 2010

A039621 Triangle of Lehmer-Comtet numbers of 2nd kind.

Original entry on oeis.org

1, -1, 1, 4, -3, 1, -27, 19, -6, 1, 256, -175, 55, -10, 1, -3125, 2101, -660, 125, -15, 1, 46656, -31031, 9751, -1890, 245, -21, 1, -823543, 543607, -170898, 33621, -4550, 434, -28, 1, 16777216, -11012415, 3463615, -688506, 95781, -9702, 714, -36, 1

Offset: 1

Len Smiley

Also the Bell transform of (-n)^n adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 16 2016

			The triangle T(n, k) begins:
[1]       1;
[2]      -1,      1;
[3]       4,     -3,       1;
[4]     -27,     19,      -6,     1;
[5]     256,   -175,      55,   -10,     1;
[6]   -3125,   2101,    -660,   125,   -15,   1;
[7]   46656, -31031,    9751, -1890,   245, -21,   1;
[8] -823543, 543607, -170898, 33621, -4550, 434, -28, 1;

D. H. Lehmer, Numbers Associated with Stirling Numbers and x^x, Rocky Mountain J. Math., 15(2) 1985, pp. 461-475.

A008296 (matrix inverse), A354794 (variant), A045531 (column |a(n, 2)|).

Cf. A185164.

R := proc(n, k, m) option remember;
   if k < 0  or n < 0 then 0 elif k = 0 then 1 else
   m*R(n, k-1, m) + R(n-1, k, m+1) fi end:
A039621 := (n, k) -> (-1)^(n-k)*R(k-1, n-k, n-k):
seq(seq(A039621(n, k), k = 1..n), n = 1..9); # Peter Luschny, Jun 10 2022 after Vladimir Kruchinin

a[1, 1] = 1; a[n_, k_] := 1/(k-1)! Sum[((-1)^(n-k-i)*Binomial[k-1, i]*(n-i-1)^(n-1)), {i, 0, k-1}];
Table[a[n, k], {n, 1, 10}, {k, 1, n}]//Flatten (* Jean-François Alcover, Jun 03 2019 *)

T(n,k,m):=if k<0 or n<0 then 0 else if k=0 then 1 else m*T(n,k-1,m)+T(n-1,k,m+1);
a(n,k):=if nVladimir Kruchinin, Mar 07 2020

tabl(nn) = {for (n = 1, nn, for (k = 1, n, print1(sum(i = 0, k-1,(-1)^(n-k-i)*binomial(k-1, i)*(n-i-1)^(n-1))/(k-1)!, ", ");); print(););} \\ Michel Marcus, Aug 28 2013

# uses[bell_matrix from A264428]
# Adds 1,0,0,0,... as column 0 at the left side of the triangle.
bell_matrix(lambda n: (-n)^n, 7) # Peter Luschny, Jan 16 2016

(k-1)!*a(n, k) = Sum_{i=0..k-1}((-1)^(n-k-i)*binomial(k-1, i)*(n-i-1)^(n-1)).

a(n,k) = (-1)^(n-k)*T(k,n-k,n-k), n>=k, where T(n,k,m)=m*T(n,m-1,k)+T(n-1,k,m+1), T(n,0,m)=1. - Vladimir Kruchinin, Mar 07 2020

cp's OEIS Frontend

A045406 A diagonal of A008296.

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A059302 A diagonal of A008296.

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A298511 Central Lehmer-Comtet numbers of the first kind: a(n) = A008296(2n,n).

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A345651 Fourth column of A008296.

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A347276 Third column of A008296.

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A039620 Erroneous version of A008296.

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A008297 Triangle of Lah numbers.

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A005727 n-th derivative of x^x at x=1. Also called Lehmer-Comtet numbers.