A006153 -id:A006153 - cp's OEIS frontend

A161633 E.g.f. satisfies A(x) = 1/(1 - xexp(xA(x))).

1, 1, 4, 27, 268, 3525, 57966, 1146061, 26500552, 702069129, 20974309210, 697754762001, 25584428686620, 1025230366195789, 44579963354153878, 2090676600895922565, 105191995364927688976, 5652501986238910061073, 323083811850594613809714, 19573120681427758058921881

Offset: 0

Paul D. Hanna, Jun 18 2009

nonn

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 268*x^4/4! + 3525*x^5/5! +...
exp(x*A(x)) = 1 + x + 3*x^2/2! + 19*x^3/3! + 181*x^4/4! + 2321*x^5/5! +...

Seiichi Manyama, Table of n, a(n) for n = 0..374

Cf. A006153, A161630 (e.g.f. = exp(x*A(x))), A213644, A364980, A364981.

Cf. A366232, A366233, A366234, A366235.

Flatten[{1,Table[n!*Sum[Binomial[n+1,k]/(n+1) * k^(n-k)/(n-k)!,{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jan 10 2014 *)

a(n,m=1)=n!*sum(k=0,n,binomial(n+m,k)*m/(n+m)*k^(n-k)/(n-k)!)

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = 1 + x*A(x)*exp(x*A(x)).
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(x)) ).
(3) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-1)*x*A(x)) for all fixed nonnegative m.
a(n) = n! * Sum_{k=0..n} binomial(n+1,k)/(n+1) * k^(n-k)/(n-k)!.
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! then a(n,m) = n! * Sum_{k=0..n} binomial(n+m,k)*m/(n+m) * k^(n-k)/(n-k)!.
a(n) ~ n^(n-1) * c * ((c-1)*c)^(n+1/2) / (sqrt(2*c-1) * exp(n)), where c = 1 + 1/(2*LambertW(1/2)) = 2.4215299358831166... - Vaclav Kotesovec, Jan 10 2014

A213644 E.g.f. satisfies: A(x) = 1 + xA(x)^2exp(x*A(x)).

Original entry on oeis.org

1, 1, 6, 63, 988, 20725, 546246, 17364445, 646910328, 27652214313, 1334291800330, 71749167806041, 4255000637001588, 275904038948566093, 19420072633921942542, 1474700254793433800805, 120174737260376219862256, 10461031446553525766071249, 968785652772129485926955538

Offset: 0

Paul D. Hanna, Jun 17 2012

nonn

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 63*x^3/3! + 988*x^4/4! + 20725*x^5/5! +...
such that A(x-x^2*exp(x)) = 1/(1-x*exp(x)) where:
1/(1-x*exp(x)) = 1 + x + 4*x^2/2! + 21*x^3/3! + 148*x^4/4! + 1305*x^5/5! +...+ A006153(n)*x^n/n! +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A213643, A006153, A161633.

Flatten[{1,Table[1/(n+1)*Sum[k^(n-k)/(n-k)!*(n+k)!/k!,{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jul 15 2013 *)

{a(n)=1/(n+1)*sum(k=0, n, k^(n-k)/(n-k)! * (n+k)!/k! )}

{a(n)=n!*polcoeff((1/x)*serreverse(x-x^2*exp(x+x*O(x^n))),n)}
for(n=0,25,print1(a(n),", "))

E.g.f. satisfies: A(x) = 1/(1 - x*A(x)*exp(x*A(x))).
E.g.f. satisfies: A(x-x^2*exp(x)) = 1/(1-x*exp(x)).
a(n) = 1/(n+1) * Sum_{k=0..n} k^(n-k)/(n-k)! * (n+k)!/k!.
a(n) = A213643(n+1)/(n+1).
Limit n->infinity (a(n)/n!)^(1/n) = r*(1+r)/(1-r) = 5.5854662015218413..., where r = 0.7603592340333989... is the root of the equation (1-r^2)/r^2 = exp((r-1)/r), same as for A213643. - Vaclav Kotesovec, Jul 15 2013
a(n) ~ (1+r)*sqrt(r/(1+2*r-r^2)) * n^(n-1) * (r*(1+r)/(1-r))^n / exp(n). - Vaclav Kotesovec, Dec 28 2013

A052848 Number of ordered set partitions with a designated element in each block and no block containing less than two elements.

Original entry on oeis.org

1, 0, 2, 3, 28, 125, 1146, 8827, 94200, 1007001, 12814390, 172114151, 2584755636, 41436880069, 721702509906, 13397081295795, 266105607506416, 5605474012933169, 125164378600050798, 2948082261121889983, 73122068527848758700, 1903894649651935410141

Offset: 0

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

a(n) is the number of functional digraphs on {1,2,...,n} such that no node is at a distance greater than one from a cycle (A006153) and every recurrent element has at least one nonrecurrent element mapped to it. - Geoffrey Critzer, Dec 07 2012

Alois P. Heinz, Table of n, a(n) for n = 0..434
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 816

Cf. A000296.

spec := [S,{B=Prod(Z,C),C=Set(Z,1 <= card),S=Sequence(B)},labeled]: seq(combstruct[count](spec, size=n), n=0..20);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-j)*binomial(n, j)*j, j=2..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, May 11 2016

nn=20; a=x Exp[x]; First[Range[0,nn]! CoefficientList[Series[1/(1-x (Exp[x]-1+y)), {x,0,nn}], {y,x}]] Range[0,nn]! (* Geoffrey Critzer, Dec 07 2012 *)

a(n):=n!*sum((k!*stirling2(n-k,k))/(n-k)!,k,0,n/2); /* Vladimir Kruchinin, Nov 16 2011 */

E.g.f.: -1/(-1+x*exp(x)-x).
a(n) = n!*Sum_{k=0..floor(n/2)} k!*Stirling2(n-k,k)/(n-k)!. - Vladimir Kruchinin, Nov 16 2011
a(n) ~ n!/(1+r+r^2) * r^(n+2), where r = 1.23997788765655... is the root of the equation log(1+r)=1/r. - Vaclav Kotesovec, Oct 05 2013
a(0) = 1; a(n) = n * Sum_{k=2..n} binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Dec 04 2023

Better name from Geoffrey Critzer, Dec 10 2012

A072597 Expansion of 1/(exp(-x) - x) as exponential generating function.

Original entry on oeis.org

1, 2, 7, 37, 261, 2301, 24343, 300455, 4238153, 67255273, 1185860331, 23000296155, 486655768525, 11155073325917, 275364320099807, 7282929854486431, 205462851526617489, 6158705454187353297, 195465061563672788947, 6548320737474275229347, 230922973019493881984021

Offset: 0

Michael Somos, Jun 23 2002

Polynomials from A140749/A141412 are linked to Stirling1 (see A048594, A129841, A140749). See also P. Flajolet, X. Gourdon, B. Salvy in, available on Internet, RR-1857.pdf (preprint of unavailable Gazette des Mathematiciens 55, 1993, pp. 67-78; for graph 2 see also X. Gourdon RR-1852.pdf, pp. 64-65). What is the corresponding graph for A152650/A152656 = simplified A009998/A119502 linked, via A152818, to a(n), then Stirling2? - Paul Curtz, Dec 16 2008
Denominators in rational approximations of Lambert W(1). See Ramanujan, Notebooks, volume 2, page 22: "2. If e^{-x} = x, shew that the convergents to x are 1/2, 4/7, 21/37, 148/261, &c." Numerators in A006153. - Michael Somos, Jan 21 2019
Call an element g in a semigroup a group element if g^j = g for some j > 1. Then a(n) is the number of group elements in the semigroup of partial transformations of an n-set. Hence a(n) = Sum_{k=0..n} A154372(n,k)*k!. - Geoffrey Critzer, Nov 27 2021

			G.f. = 1 + 2*x + 7*x^2 + 37*x^3 + 261*x^4 + 2301*x^5 + 24343*x^6 + ...

O. Ganyushkin and V Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009, page 70.
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 2, see page 22.

Seiichi Manyama, Table of n, a(n) for n = 0..411
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 March 2013.
G. Jiraskova and J. Shallit, The state complexity of star-complement-star, arXiv preprint arXiv:1203.5353 [cs.FL], 2012. - From _N. J. A. Sloane_, Sep 21 2012

Cf. A000110, A006153, A030178, A089148.
Cf. A048993, A052820.
Cf. A006153, A154372.

CoefficientList[Series[1/(Exp[-x]-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 26 2013 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ 1 / (Exp[-x] - x), {x, 0, n}]]; (* Michael Somos, Jan 21 2019 *)
a[ n_] := If[ n < 0, 0, n! Sum[ (n - k + 1)^k / k!, {k, 0, n}]]; (* Michael Somos, Jan 21 2019 *)

{a(n) = if( n<0, 0, n! * polcoeff( 1 / (exp(-x + x * O(x^n)) - x), n))};

{a(n) = if( n<0, 0, n! * sum(k=0, n, (n-k+1)^k / k!))}; /* Michael Somos, Jan 21 2019 */

E.g.f.: 1 / (exp(-x) - x).
a(n) = n!*Sum_{k=0..n} (n-k+1)^k/k!. - Vladeta Jovovic, Aug 31 2003
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*A052820(k). - Vladeta Jovovic, Apr 12 2004
Recurrence: a(n+1) = 1 + Sum_{j=1..n} binomial(n, j)*a(j)*j. - Jon Perry, Apr 25 2005
E.g.f.: 1/(Q(0) - x) where Q(k) = 1 - x/(2*k+1 - x*(2*k+1)/(x - (2*k+2)/Q(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 04 2013
a(n) ~ n!/((1+c)*c^(n+1)), where c = A030178 = LambertW(1) = 0.5671432904... - Vaclav Kotesovec, Jun 26 2013
O.g.f.: Sum_{k>=0} k!*x^k/(1 - (k + 1)*x)^(k+1). - Ilya Gutkovskiy, Oct 09 2018
a(n) = A006153(n+1)/(n+1). - Seiichi Manyama, Nov 05 2024

A089148 Expansion of e.g.f.: 1/(exp(x) - x).

Original entry on oeis.org

1, 0, -1, -1, 5, 19, -41, -519, -183, 19223, 73451, -847067, -8554547, 32488611, 977198559, 1325135969, -116987762287, -860498433233, 13730866757587, 243612350234973, -1120827248102379, -62079344419449925, -185852602587850681, 15185914155303053209

Offset: 0

Wouter Meeussen, Dec 06 2003

sign

INVERTi transform of [1, 1, 1/2, 1/6, 1/24, 1/120, ...] = [1, 0, -1/2, 1/6, 5/24, -19/120, -41/720, 519/5040, -183/40320, -19223/362880, ...]. - Gary W. Adamson, Oct 08 2008

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A006153, A302397, A089087.

b:= proc(n) b(n):= -`if` (n<0, 1, add(b(n-i)/(i-1)!, i=1..n+1)) end:
a:= n-> (-1)^n*n!*b(n):
seq(a(n), n=0..30);  # Alois P. Heinz, May 29 2013

a = CoefficientList[Series[1/( E^ x - x), {x, 0, 30}], x]; Table[(n - 1)! *a[[n]], {n, 1, Length[a]}] (* Roger L. Bagula and Gary W. Adamson, Oct 06 2008 *)
With[{nn=30},CoefficientList[Series[1/(Exp[x]-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 03 2017 *)

a(n):=sum(sum(k!*binomial(n,l)*(-1)^(k-l)*stirling2(n-l,k-l), l,0,k), k,0,n); /* Vladimir Kruchinin, May 29 2013 */

a(n):=n!*sum((-n-1+k)^k/k!,k,0,n); /* Tani Akinari, Mar 26 2023 */

x='x+O('x^66); Vec(serlaplace(1/(exp(x)-x))) \\ Joerg Arndt, May 29 2013

def A089148_list(len):
    f, R, C = 1, [], [1]+[0]*len
    for n in (1..len):
        for k in range(n, 0, -1):
            C[k] = C[k-1]*(1/(k-1) if k>1 else 1)
        C[0] = -sum((-1)^k*C[k] for k in (1..n))
        R.append(C[0]*f)
        f *= n
    return R
print(A089148_list(24)) # Peter Luschny, Feb 21 2016

E.g.f.: -(1+1/(G(0)-1))/x where G(k) = 1 - (k+1)/(1 - x/(x + (k+1)^2/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 07 2012
a(n) = Sum_{k=0..n} Sum_{m=0..k} k!*binomial(n,m)*(-1)^(k-m)*Stirling2(n-m,k-m). - Vladimir Kruchinin, May 29 2013
Lim sup n->oo |a(n)/n!|^(1/n) = 1/abs(LambertW(-1)) = 0.727507111152... - Vaclav Kotesovec, Aug 13 2013
a(0) = 1; a(n) = -Sum_{k=2..n} binomial(n,k) * a(n-k). - Ilya Gutkovskiy, Feb 09 2020
a(n) = n!*Sum_{k=0..n} (-n-1+k)^k/k!. - Tani Akinari, Mar 25 2023
a(n) = Sum_{k=0..n} A089087(n, k). - Peter Luschny, Mar 25 2023

A302397 Expansion of e.g.f. 1/(1 + x*exp(x)).

Original entry on oeis.org

1, -1, 0, 3, -4, -25, 114, 287, -4152, 1647, 192230, -807961, -10164804, 111209111, 454840554, -14657978385, 21202175504, 1988791958879, -15488971798194, -260886468394153, 4872247004699460, 23537372210149959, -1365745577227898350, 4274609859520565663, 364461939727273277016

Offset: 0

Ilya Gutkovskiy, Apr 07 2018

sign

			1/(1 + x*exp(x)) = 1 - x/1! + 3*x^3/3! - 4*x^4/4! - 25*x^5/5! + 114*x^6/6! + 287*x^7/7! - 4152*x^8/8! + 1647*x^9/9! + ...

Seiichi Manyama, Table of n, a(n) for n = 0..474

Cf. A006153, A009306, A089148.

a:=series(1/(1+x*exp(x)),x=0,25): seq(n!*coeff(a,x,n),n=0..24); # Paolo P. Lava, Mar 26 2019

nmax = 24; CoefficientList[Series[1/(1 + x Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[n! Sum[(-1)^(n - k) (n - k)^k/k!, {k, 0, n}], {n, 24}]]
Join[{1}, Table[Sum[(-1)^k k! k^(n - k) Binomial[n, k], {k, 0, n}], {n, 24}]]

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

A216794 Number of set partitions of {1,2,...,n} with labeled blocks and a (possibly empty) subset of designated elements in each block.

Original entry on oeis.org

1, 2, 12, 104, 1200, 17312, 299712, 6053504, 139733760, 3628677632, 104701504512, 3323151509504, 115063060869120, 4316023589937152, 174347763227738112, 7545919601962287104, 348366745238330081280, 17087957176042900815872, 887497598764802460352512

Offset: 0

Geoffrey Critzer, Sep 16 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200
José A. Adell, Beáta Bényi, Venkat Murali, and Sithembele Nkonkobe, Generalized Barred Preferential Arrangements, Transactions on Combinatorics (2022).
Sithembele Nkonkobe, Venkat Murali, and Béata Bényi, Generalised Barred Preferential Arrangements, arXiv:1907.08944 [math.CO], 2019.
Eric Weisstein's World of Mathematics, Polylogarithm.

Cf. A000556, A000670, A006153, A055882, A080253.

a := n -> 2^(n-1)*(polylog(-n, 1/2)+`if`(n=0,1,0)):
seq(round(evalf(a(n),32)), n=0..18); # Peter Luschny, Nov 03 2015
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n, j)*2^j, j=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 04 2019

nn=25;a=Exp[2x]-1;Range[0,nn]!CoefficientList[Series[1/(1-a),{x,0,nn}],x]
Round@Table[(-1)^(n+1) (PolyLog[-n, Sqrt[2]] + PolyLog[-n, -Sqrt[2]])/4, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 31 2015 *)

a(n) = 2^(n-1)*(polylog(-n, 1/2) + 0^n); \\ Michel Marcus, May 30 2018

def A216794(n):
    return 2^n*add(add((-1)^(j-i)*binomial(j,i)*i^n for i in range(n+1)) for j in range(n+1))
[A216794(n) for n in range(18)] # Peter Luschny, Jul 22 2014

E.g.f.: 1/(2 - exp(2*x)).
E.g.f.: 1 + 2*x/(G(0) - 2*x) where G(k) = 2*k+1 - x*2*(2*k+1)/(2*x + (2*k+2)/(1 + 2*x/G(k+1))); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 26 2012
E.g.f.: 1 + 2*x/( G(0) - 2*x ) where G(k) = 1 - 2*x/(1 + (1*k+1)/G(k+1)); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 02 2013
G.f.: 1/G(0) where G(k) = 1 - x*(2*k+2)/( 1 - 4*x*(k+1)/G(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 23 2013
a(n) ~ n! * (2/log(2))^n/log(4). - Vaclav Kotesovec, Sep 24 2013
G.f.: T(0)/(1-2*x), where T(k) = 1 - 8*x^2*(k+1)^2/( 8*x^2*(k+1)^2 - (1-2*x-6*x*k)*(1-8*x-6*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 14 2013
From Vladimir Reshetnikov, Oct 31 2015: (Start)
a(n) = (-1)^(n+1)*(Li_{-n}(sqrt(2)) + Li_{-n}(-sqrt(2)))/4, where Li_n(x) is the polylogarithm.
Li_{-n}(sqrt(2)) = (-1)^(n+1)*(2*a(n) + A080253(n)*sqrt(2)).
(End)
a(n) = 2^(n-1)*(Li_{-n}(1/2) + 0^n) with 0^0=1. - Peter Luschny, Nov 03 2015
From Peter Bala, Oct 18 2023: (Start)
a(n) = 2^n * A000670(n)
Inverse binomial transform of A080253.
The sequence is the first column of the array (2*I - P^2)^(-1), where P denotes Pascal's triangle A007318. (End)

A354436 a(n) = n! * Sum_{k=0..n} k^(n-k)/k!.

Original entry on oeis.org

1, 1, 3, 13, 85, 801, 10231, 168253, 3437673, 85162465, 2511412651, 86805640461, 3469622549053, 158523442439233, 8198514736542495, 476003264246418301, 30804251925861439441, 2207978115389469465153, 174304316334466458575443

Offset: 0

Seiichi Manyama, May 28 2022

nonn

Cf. A006153, A026898, A010844, A277452, A277506, A354437.

Join[{1}, Table[n!*Sum[k^(n-k)/k!, {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, May 28 2022 *)

PARI
```
a(n) = n!*sum(k=0, n, k^(n-k)/k!);
```

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1-k*x)))))

from math import factorial
def A354436(n): return sum(factorial(n)*k**(n-k)//factorial(k) for k in range(n+1)) # Chai Wah Wu, May 28 2022

E.g.f.: Sum_{k>=0} x^k / (k! * (1 - k*x)).
a(n) ~ sqrt(Pi) * exp((2*n-1)/(2*LambertW(exp(1/2)*(2*n-1)/4)) - 2*n) * n^(2*n + 1/2) / (sqrt(1 + LambertW(exp(1/2)*(2*n-1)/4)) * 2^n * LambertW(exp(1/2)*(2*n-1)/4)^n). - Vaclav Kotesovec, May 28 2022
a(n) = Sum_{k=0..n} (n-k)^k*k!*binomial(n,k). - Ridouane Oudra, Jun 17 2025

A193421 E.g.f.: Sum_{n>=0} x^n * exp(n^2*x).

Original entry on oeis.org

1, 1, 4, 33, 436, 8185, 206046, 6622945, 263313688, 12627149265, 716160702970, 47284266221401, 3587061106583604, 309251317536586633, 30017652739792964806, 3254137305364883664945, 391238883136463492841136, 51846176797206158144925985

Offset: 0

Paul D. Hanna, Jul 27 2011

nonn

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 436*x^4/4! + 8185*x^5/5! + 206046*x^6/6! +...
where
A(x) = 1 + x*exp(x) + x^2*exp(4*x) + x^3*exp(9*x) + x^4*exp(16*x) +...
By a q-series identity:
A(x) = 1 + x*exp(x)*(1-x*exp(x))/(1-x*exp(3*x)) + x^2*exp(2*x)*(1-x*exp(x))*(1-x*exp(5*x))/((1-x*exp(3*x))*(1-x*exp(7*x))) + x^3*exp(3*x)*(1-x*exp(x))*(1-x*exp(5*x))*(1-x*exp(9*x))/((1-x*exp(3*x))*(1-x*exp(7*x))*(1-x*exp(11*x))) +...

Seiichi Manyama, Table of n, a(n) for n = 0..290

Cf. A006153, A193467.

Flatten[{1,Table[n! * Sum[(n-k)^(2*k)/k!,{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Oct 21 2014 *)

{a(n)=local(Egf); Egf=sum(m=0, n, x^m*exp(m^2*x+x*O(x^n))); n!*polcoeff(Egf, n)}

/* q-series identity: */
{a(n)=local(A=1+x);for(i=1, n, A=sum(m=0, n, x^m*exp(m*x+x*O(x^n))*prod(k=1, m, (1-x*exp((4*k-3)*x+x*O(x^n)))/(1-x*exp((4*k-1)*x+x*O(x^n)))))); n!*polcoeff(A, n)}

{a(n) = n!*sum(k=0,n, (n-k)^(2*k)/k!)}
for(n=0,20,print1(a(n),", "))

E.g.f.: A(x) = Sum_{n>=0} x^n*exp(n*x)*Product_{k=1..n} (1 - x*exp((4*k-3)*x)) / (1 - x*exp((4*k-1)*x)), due to a q-series identity.
Let q = exp(x), then the e.g.f. equals the continued fraction:
A(x) = 1/(1- q*x/(1- q*(q^2-1)*x/(1- q^5*x/(1- q^3*(q^4-1)*x/(1- q^9*x/(1- q^5*(q^6-1)*x/(1- q^13*x/(1- q^7*(q^8-1)*x/(1- ...))))))))), due to a partial elliptic theta function identity.
a(n) = n! * Sum_{k=0..n} (n-k)^(2*k)/k!. - Paul D. Hanna, Jan 19 2013
O.g.f.: Sum_{k>=0} k! * x^k / (1 - k^2*x)^(k+1). - Ilya Gutkovskiy, Jul 02 2019
log(a(n)) ~ n*(2*(log(n) - 1) + LambertW(sqrt(n))*(3*log(n) - 2*log(1 + LambertW(sqrt(n))) + 2*LambertW(sqrt(n)))) / (2*(1 + LambertW(sqrt(n)))). - Vaclav Kotesovec, Nov 26 2022

A009444 E.g.f. log(1 + x*exp(-x)).

Original entry on oeis.org

0, 1, -3, 11, -58, 409, -3606, 38149, -470856, 6641793, -105398650, 1858413061, -36044759796, 762659322385, -17481598316742, 431535346662645, -11413394655983536, 321989729198400385, -9651573930139850610

Offset: 0

R. H. Hardin

abs(a(n)) is the number of connected functions f:{1,2,...,n}->{1,2,...,n} such that every element is mapped into a recurrent element. Cf. A006153. - Geoffrey Critzer, May 24 2012

G. C. Greubel, Table of n, a(n) for n = 0..400
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 125

Cf. A006153, A009306, A305990.

With[{nmax = 40}, CoefficientList[Series[Log[1 + x*Exp[-x]], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Nov 22 2017 *)

a(n):=(-1)^(n+1)*n!*sum(m^(n-m-1)/(n-m)!,m,1,n); /* Vladimir Kruchinin, Oct 08 2011 */

x='x+O('x^66); /* that many terms */
egf=1/(1+x/exp(x)); /* = 1 - x + 2*x^2 - 7/2*x^3 + 37/6*x^4 - 87/8*x^5 +... */
Vec(serlaplace(egf)) /* show terms */ /* Joerg Arndt, Apr 30 2011 */

A009444 = lambda n: (-1)^(n+1)*factorial(n)*sum(m^(n-m-1)/factorial(n-m) for m in (1..n))
[A009444(n) for n in (0..9)] # Peter Luschny, Jan 18 2016

abs(a(n)) is asymptotic to (n-1)!/LambertW(1)^n. - Vladeta Jovovic, Jul 12 2007
Sequence of absolute values has e.g.f. log(1/(1-x*exp(x))). - Joerg Arndt, Apr 30 2011
a(n) = (-1)^(n+1)*n!*sum(m=1..n, m^(n-m-1)/(n-m)!). - Vladimir Kruchinin, Oct 08 2011
a(n) = (-1)^(n + 1) * n + Sum_{k=1..n-1} (-1)^(n - k) * binomial(n-1,k-1) * (n - k) * a(k). - Ilya Gutkovskiy, Jan 17 2020

Extended with signs by Olivier Gérard, Mar 15 1997

Definition corrected by Joerg Arndt, Apr 30 2011

cp's OEIS Frontend

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A089148 Expansion of e.g.f.: 1/(exp(x) - x).

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A302397 Expansion of e.g.f. 1/(1 + x*exp(x)).

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A161633 E.g.f. satisfies A(x) = 1/(1 - xexp(xA(x))).

A213644 E.g.f. satisfies: A(x) = 1 + xA(x)^2exp(x*A(x)).