A216688 -id:A216688 - cp's OEIS frontend

A053530 Expansion of e.g.f.: exp(-x - x^2/2 + x*exp(x)).

1, 0, 1, 3, 7, 35, 171, 847, 5041, 32643, 223705, 1659581, 13182159, 110802133, 984241363, 9212696235, 90477239521, 929604133343, 9969157068273, 111329454692485, 1291932988047775, 15550838026589061, 193833398512358011, 2498039016973836491

Offset: 0

N. J. A. Sloane, Jan 16 2000

nonn

The number of simple labeled graphs on n nodes whose connected components are stars. - Geoffrey Critzer, Dec 10 2011

Equivalently, the number of minimal edge covers of the complete graph K_n. - Andrew Howroyd, Aug 04 2017

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.15(b).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
Eric Weisstein's World of Mathematics, Complete Graph
Eric Weisstein's World of Mathematics, Minimal Edge Cover

Cf. A000248, A210655.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(-x -x^2/2 +x*Exp(x)) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 15 2019

nn = 30; a = x Exp[x]; Range[0, nn]! CoefficientList[Series[Exp[a - x^2/2! - x], {x, 0, nn}], x] (* Geoffrey Critzer, Dec 10 2011 *)
CoefficientList[Series[Exp[-x - x^2/2 + x Exp[x]], {x, 0, 30}], x] Range[0, 30]! (* Eric W. Weisstein, Aug 10 2017 *)
Table[n! Sum[1/k! (Binomial[k, n-k] 2^(k-n) (-1)^k + Sum[Binomial[k, j] Sum[j^(i-j)/(i-j)! Binomial[k-j, n-i-k+j] 2^(i-j+k-n) (-1)^(k-j), {i, j, n-k+j}], {j, k}]), {k, n}], {n, 30}] (* Eric W. Weisstein, Aug 10 2017 *)

a(n):=n!*sum((binomial(k,n-k)*2^(k-n)*(-1)^k +sum(binomial(k,j) *sum(j^(i-j)/(i-j)!*binomial(k-j,n-i-k+j)*(1/2)^(n-i-k+j)*(-1)^(k-j),i,j,n-k+j),j,1,k))/k!,k,1,n); /* Vladimir Kruchinin, Sep 10 2010 */

x='x+O('x^30); Vec(serlaplace(exp(-x-1/2*x^2+x*exp(x)))) \\ Altug Alkan, Aug 10 2017

m = 30; T = taylor(exp(-x -x^2/2 +x*exp(x)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 15 2019

a(n) = n!*Sum_{k=1..n} (1/k!)*( binomial(k, n-k)*2^(k-n)*(-1)^k + Sum_{j=1..k} binomial(k,j)* (Sum_{i=j..n-k+j} (j^(i-j)/(i-j)! * binomial(k-j,n-i-k+j)*(1/2)^(n-i-k+j)*(-1)^(k-j)) ) ), n>0. - Vladimir Kruchinin, Sep 10 2010
From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ n^n / (exp(r^2/2 + n*r/(1+r)) * r^n * sqrt(r^2*(1+r)/n + 2+r-1/(1+r))), where r is the root of the equation r*(exp(r)*(1+r)-1-r) = n.
(a(n)/n!)^(1/n) ~ exp(1/(2*LambertW(sqrt(n)/2)))/(2*LambertW(sqrt(n)/2)).
(End)

A052506 Expansion of e.g.f. exp(x*exp(x)-x).

Original entry on oeis.org

1, 0, 2, 3, 16, 65, 336, 1897, 11824, 80145, 586000, 4588001, 38239224, 337611001, 3144297352, 30779387745, 315689119456, 3383159052833, 37790736663456, 439036039824193, 5294386116882280, 66155074120062921, 855156188538926296, 11418964004032623809

Offset: 0

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

a(n) is the number of forests of rooted labeled trees with height exactly one. Equivalently, the number of idempotent mappings from {1,2,...,n} into {1,2,...,n} where each fixed point has at least one (other than itself) element mapped to it. See the second summation formula provided by Vladeta Jovovic with conditions on k, the number of fixed points. - Geoffrey Critzer, Sep 20 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 39
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Cf. A000248, A240989.

Cf. A351736, A351737.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x*Exp(x)-x) )); [Factorial(n-1)*b[n]: n in [1..m-1]]; // G. C. Greubel, May 13 2019

spec := [S,{S=Set(Tree), Tree=Prod(Z,Set(Z,0 < card))},labeled]: seq(combstruct[count](spec, size=n), n=0..20);

nn=20;Range[0,nn]! CoefficientList[Series[Exp[x(Exp[x]-1)], {x,0,nn}], x]  (* Geoffrey Critzer, Sep 20 2012 *)

my(x='x+O('x^30)); Vec(serlaplace( exp(x*exp(x)-x) )) \\ G. C. Greubel, Nov 15 2017

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(k-1)*x)^(k+1))) \\ Seiichi Manyama, Aug 29 2022

m = 30; T = taylor(exp(x*exp(x)-x), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 13 2019

a(n) = Sum_{k=0..n} binomial(n, k)*(n-k-1)^k. - Vladeta Jovovic, Apr 12 2003
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*k!*Stirling2(n-k, k). - Vladeta Jovovic, Dec 19 2004
a(n) ~ exp((1-r*(n+r))/(1+r)) * n^(n+1/2) * sqrt(1+r) / (r^n * sqrt((1+r)^3 + n*(1+3*r+r^2))), where r satisfies exp(r)*(1+r) - (1+n)/r = 1. - Vaclav Kotesovec, Aug 04 2014
(a(n)/n!)^(1/n) ~ exp(1/(2*LambertW(sqrt(n)/2))) / (2*LambertW(sqrt(n)/2)). - Vaclav Kotesovec, Aug 06 2014
G.f.: Sum_{k>=0} x^k / (1 - (k-1)*x)^(k+1). - Seiichi Manyama, Aug 29 2022

A003727 Expansion of e.g.f. exp(x * cosh(x)).

Original entry on oeis.org

1, 1, 1, 4, 13, 36, 181, 848, 3865, 23824, 140521, 871872, 6324517, 44942912, 344747677, 2860930816, 23853473329, 213856723200, 1996865965009, 19099352929280, 193406280000061, 2010469524579328, 21615227339380357, 242177953175506944

Offset: 0

R. H. Hardin

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..500
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Cf. A009233, A191509.

m:=50; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x*Cosh(x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Sep 09 2018

CoefficientList[Series[E^(x*Cosh[x]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 05 2014 *)
Table[Sum[BellY[n, k, Mod[Range[n], 2] Range[n]], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

a(n):=sum(if n=k then n! else 1/2^k*sum(binomial(n,k)*binomial(k,i)*(k-2*i)^(n-k),i,0,k),k,1,n); /* Vladimir Kruchinin, Aug 22 2010 */

x='x+O('x^66);
Vec(serlaplace(exp( x * cosh(x) )))
/* Joerg Arndt, Sep 14 2012 */

a(n) = Sum_{k=1..n} (if n=k then n! otherwise (1/2)^k*Sum_{i=0..k} binomial(n,k)* binomial(k,i)*(k-2*i)^(n-k)), n>0. - Vladimir Kruchinin, Aug 22 2010
a(n) ~ exp(r*cosh(r)-n) * n^n / (r^n * sqrt(3+(r*(r^2-2)*cosh(r))/n)), where r is the root of the equation r*(cosh(r)+r*sinh(r)) = n. - Vaclav Kotesovec, Aug 05 2014
a(n)^(1/n) ~ n*exp(1/(2*LambertW(sqrt(n/2)))-1) / (2*LambertW(sqrt(n/2))). - Vaclav Kotesovec, Aug 05 2014
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * (2*k+1) * a(n-2*k-1). - Ilya Gutkovskiy, Feb 24 2022

Extended and formatted by Olivier Gérard, Mar 15 1997

A216507 E.g.f. exp( x^2 * exp(x) ).

Original entry on oeis.org

1, 0, 2, 6, 24, 140, 870, 5922, 45416, 381096, 3442410, 33382910, 345803172, 3801763836, 44156760830, 539962736250, 6929042527920, 93032248209872, 1303556965679826, 19018807375195638, 288341417011487420, 4534168069704168420, 73829219253218066022, 1242905562198878544626

Offset: 0

Joerg Arndt, Sep 14 2012

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..516 (terms 0..200 from Vincenzo Librandi)
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Column k=2 of A292978.
Cf. A216688 (e.g.f. exp(x*exp(x^2))), A216689 (e.g.f. exp(x*exp(x)^2)).
Cf. A000248 (e.g.f. exp(x*exp(x))), A003725 (e.g.f. exp(x*exp(-x))).

With[{nn = 25}, CoefficientList[Series[Exp[x^2 Exp[x]], {x, 0, nn}],
   x] Range[0, nn]!] (* Bruno Berselli, Sep 14 2012 *)

x='x+O('x^66);
Vec(serlaplace(exp( x^2 * exp(x) )))
/* Joerg Arndt, Sep 14 2012 */

From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ n^n / (exp(n*(1+r)/(2+r)) * r^n * sqrt((1+r)*(4+r)/(2+r))), where r is the root of the equation r^2*(2+r)*exp(r) = n.
(a(n)/n!)^(1/n) ~ exp(1/(3*LambertW(n^(1/3)/3))) / (3*LambertW(n^(1/3)/3)).
(End)
a(n) = Sum_{k = 0..n/2} C(n,2*k) * ((2*k)!/k!) * k^(n-2*k). - David Einstein, Oct 30 2016

A065143 a(n) = Sum_{k=0..n} Stirling2(n,k)(1+(-1)^k)2^k/2.

Original entry on oeis.org

1, 0, 4, 12, 44, 220, 1228, 7196, 45004, 303900, 2201676, 16920860, 136966860, 1163989788, 10364408140, 96463232284, 935872773068, 9440653262620, 98809201693260, 1071131795708188, 12007932126074060

Offset: 0

Karol A. Penson, Oct 17 2001

nonn

Stirling transform of A199572 (aerated powers of 4).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function
Eric Weisstein's MathWorld, Bell Polynomial.

Column k=4 of A357681.

Cf. A199572, A264036, A357598.

Table[Sum[StirlingS2[n,k]*(1+(-1)^k)*2^k/2,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Aug 06 2014 *)
Table[(BellB[n, 2] + BellB[n, -2])/2, {n, 0, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)

a(n) = sum(k=0, n, stirling(n,k,2)*(1+(-1)^k)*2^k/2); \\ Michel Marcus, Nov 02 2015

x='x+O('x^50); Vec(serlaplace(cosh(2*exp(x)-2))) \\ G. C. Greubel, Nov 16 2017

Representation as a sum of an infinite series: a(n) = exp(2)*Sum_{k = 0..infinity} ((2*k)^n*2^(2*k)/(2*k)!) - sinh(2)*sum_{k = 0..infinity}(k^n*2^k/k!), for n >= 0.
E.g.f.: cosh(2*exp(x)-2). - Vladeta Jovovic, Sep 14 2003
From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ n^n * cosh(2*exp(r)-2) / (r^n * (exp(n) * sqrt(4*exp(2*r)*r^2/n + 1-n+r))), where r is the root of the equation -2*exp(r)*r*tanh(2-2*exp(r)) = n.
(a(n)/n!)^(1/n) ~ exp(1/LambertW(n/2)) / LambertW(n/2).
(End)
a(n) = (Bell_n(2) + Bell_n(-2))/2, where Bell_n(x) is n-th Bell polynomial. - Vladimir Reshetnikov, Nov 01 2015
a(n) = 1; a(n) = 4 * Sum_{k=0..n-1} binomial(n-1, k) * A357598(k). - Seiichi Manyama, Oct 12 2022

A216689 Expansion of e.g.f. exp( x * exp(x)^2 ).

Original entry on oeis.org

1, 1, 5, 25, 153, 1121, 9373, 87417, 898033, 10052353, 121492341, 1573957529, 21729801481, 318121178337, 4917743697805, 79981695655801, 1364227940101857, 24335561350365953, 452874096174214117, 8772713803852981785, 176541611843378273401, 3684142819311127955041, 79596388271096140589949

Offset: 0

Joerg Arndt, Sep 14 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Cf. A216507 (e.g.f. exp(x^2*exp(x))), A216688 (e.g.f. exp(x*exp(x^2))).
Cf. A000248 (e.g.f. exp(x*exp(x))), A003725 (e.g.f. exp(x*exp(-x))).
Cf. A240165 (e.g.f. exp(x*(1+exp(x)^2))).

With[{nn = 25}, CoefficientList[Series[Exp[x Exp[x]^2], {x, 0, nn}], x] Range[0, nn]!] (* Bruno Berselli, Sep 14 2012 *)

x='x+O('x^66);
Vec(serlaplace(exp( x * exp(x)^2 )))
/* Joerg Arndt, Sep 14 2012 */

/* From o.g.f.: */
{a(n)=local(A=1);A=sum(k=0, n, x^k/(1 - 2*k*x +x*O(x^n))^(k+1));polcoeff(A, n)}
for(n=0,25,print1(a(n),", ")) /* Paul D. Hanna, Aug 02 2014 */

/* From binomial sum: */
{a(n)=sum(k=0,n, binomial(n,k)*(2*k)^(n-k))}
for(n=0,30,print1(a(n),", ")) /* Paul D. Hanna, Aug 02 2014 */

O.g.f.: Sum_{n>=0} x^n / (1 - 2*n*x)^(n+1). - Paul D. Hanna, Aug 02 2014
a(n) = Sum_{k=0..n} binomial(n,k) * (2*k)^(n-k) for n>=0. - Paul D. Hanna, Aug 02 2014
From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ n^n / (exp(2*n*r/(1+2*r)) * r^n * sqrt((1+6*r+4*r^2)/(1+2*r))), where r is the root of the equation r*(1+2*r)*exp(2*r) = n.
(a(n)/n!)^(1/n) ~ exp(1/(2*LambertW(sqrt(n/2)))) / LambertW(sqrt(n/2)).
(End)

A358064 Expansion of e.g.f. 1/(1 - x * exp(x^2)).

Original entry on oeis.org

1, 1, 2, 12, 72, 540, 5040, 53760, 658560, 9087120, 139104000, 2343781440, 43078210560, 857676980160, 18390744852480, 422504399116800, 10353592759910400, 269576216304595200, 7431814422621388800, 216266552026593868800, 6624610236968435712000

Offset: 0

Seiichi Manyama, Oct 29 2022

Seiichi Manyama, Table of n, a(n) for n = 0..420

Cf. A216688, A358065.

With[{nn=30},CoefficientList[Series[1/(1-x Exp[x^2]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 14 2024 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x*exp(x^2))))

PARI
```
a(n) = n!*sum(k=0, n\2, (n-2*k)^k/k!);
```

a(n) = n! * Sum_{k=0..floor(n/2)} (n - 2*k)^k/k!.

a(n) ~ n! * 2^(n/2) / ((1 + LambertW(2)) * LambertW(2)^(n/2)). - Vaclav Kotesovec, Nov 01 2022

A060311 Expansion of e.g.f. exp((exp(x)-1)^2/2).

Original entry on oeis.org

1, 0, 1, 3, 10, 45, 241, 1428, 9325, 67035, 524926, 4429953, 40010785, 384853560, 3925008361, 42270555603, 478998800290, 5693742545445, 70804642315921, 918928774274028, 12419848913448565, 174467677050577515, 2542777209440690806, 38388037137038323353

Offset: 0

Vladeta Jovovic, Mar 27 2001

nonn

After the first term, this is the Stirling transform of the sequence of moments of the standard normal (or "Gaussian") probability distribution. It is not itself a moment sequence of any probability distribution. - Michael Hardy (hardy(AT)math.umn.edu), May 29 2005

a(n) is the number of simple labeled graphs on n nodes in which each component is a complete bipartite graph. - Geoffrey Critzer, Dec 03 2011

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, Ex. 3.3.5b.

Alois P. Heinz, Table of n, a(n) for n = 0..518 (first 101 terms from Harry J. Smith)
Paul Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, 17 (2014), #14.2.6.
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Column k=2 of A324162.

Cf. A052859, A330047.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
      *binomial(n-1, j-1)*Stirling2(j, 2), j=2..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 02 2019

a = Exp[x] - 1; Range[0, 20]! CoefficientList[Series[Exp[a^2/2], {x, 0, 20}], x] (* Geoffrey Critzer, Dec 03 2011 *)

a(n)=if(n<0, 0, n!*polcoeff( exp((exp(x+x*O(x^n))-1)^2/2), n)) /* Michael Somos, Jun 01 2005 */

{ for (n=0, 100, write("b060311.txt", n, " ", n!*polcoeff(exp((exp(x + x*O(x^n)) - 1)^2/2), n)); ) } \\ Harry J. Smith, Jul 03 2009

a(n) = sum(k=0, n\2, (2*k)!*stirling(n, 2*k, 2)/(2^k*k!)); \\ Seiichi Manyama, May 07 2022

E.g.f. A(x) = B(exp(x)-1) where B(x)=exp(x^2/2) is e.g.f. of A001147(2n), hence a(n) is the Stirling transform of A001147(2n). - Michael Somos, Jun 01 2005
From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ exp(1/2*(exp(r)-1)^2 - n) * n^(n+1/2) / (r^n * sqrt(exp(r)*r*(-1-r+exp(r)*(1+2*r)))), where r is the root of the equation exp(r)*(exp(r) - 1)*r = n.
(a(n)/n!)^(1/n) ~ 2*exp(1/LambertW(2*n)) / LambertW(2*n).
(End)
a(n) = Sum_{k=0..floor(n/2)} (2*k)! * Stirling2(n,2*k)/(2^k * k!). - Seiichi Manyama, May 07 2022

A240165 E.g.f.: exp( x(1 + exp(2x)) ).

Original entry on oeis.org

1, 2, 8, 44, 288, 2192, 18976, 182912, 1934848, 22231808, 275203584, 3645178880, 51370694656, 766634946560, 12066538676224, 199607631945728, 3459736006950912, 62662715180515328, 1183139425871331328, 23237689444403511296, 473852525131782946816, 10014501808427774246912

Offset: 0

Paul D. Hanna, Aug 02 2014

nonn

			E.g.f.: E(x) = 1 + 2*x + 8*x^2/2! + 44*x^3/3! + 288*x^4/4! + 2192*x^5/5! +...
where E(x) = exp(x) * exp(x*exp(2*x)).
O.g.f.: A(x) = 1 + 2*x + 8*x^2 + 44*x^3 + 288*x^4 + 2192*x^5 +...
where
A(x) = 1/(1-x) + x/(1-3*x)^2 + x^2/(1-5*x)^3 + x^3/(1-7*x)^4 + x^4/(1-9*x)^5 +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Cf. A080108, A216689.

Table[Sum[Binomial[n,k] *(2*k+1)^(n-k),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Aug 06 2014 *)
With[{nn=30},CoefficientList[Series[Exp[x(1+Exp[2x])],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jul 17 2016 *)

{a(n)=local(A=1);A=exp( x*(1 + exp(2*x +x*O(x^n))) );n!*polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

{a(n)=local(A=1);A=sum(k=0, n, x^k/(1 - (2*k+1)*x +x*O(x^n))^(k+1));polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

{a(n)=sum(k=0,n, binomial(n,k) * (2*k+1)^(n-k) )}
for(n=0,30,print1(a(n),", "))

O.g.f.: Sum_{n>=0} x^n / (1 - (2*n+1)*x)^(n+1).
a(n) = Sum_{k=0..n} binomial(n,k) * (2*k+1)^(n-k) for n>=0.
From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ exp((1+exp(2*r))*r - n) * n^(n+1/2) / (r^n * sqrt(r + exp(2*r)*r*(1 + 6*r + 4*r^2))), where r is the root of the equation r*(1 + exp(2*r) + 2*r*exp(2*r)) = n.
(a(n)/n!)^(1/n) ~ exp(1/(2*LambertW(sqrt(n/2)))) / LambertW(sqrt(n/2)).
(End)

A240989 Expansion of e.g.f. exp(x^2 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 6, 12, 20, 390, 2562, 11816, 105912, 1063530, 8815070, 81342492, 895185876, 9971185406, 112642410090, 1372455608400, 17750397057392, 236950003516626, 3286258330135734, 47688868443593540, 719345273005797900, 11222288509573985382, 181168865439054099266

Offset: 0

Vaclav Kotesovec, Aug 06 2014

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Column k=2 of A292892.
Cf. A000110, A052506, A216507, A060311.
Cf. A216688, A216689, A080108, A000248.

CoefficientList[Series[E^(x^2*(E^x-1)), {x, 0, 20}], x] * Range[0, 20]!

x='x+O('x^30); Vec(serlaplace(exp(x^2*(exp(x) - 1)))) \\ G. C. Greubel, Nov 21 2017

a(n) = n!*sum(k=0, n\3, stirling(n-2*k, k, 2)/(n-2*k)!); \\ Seiichi Manyama, Jul 09 2022

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=3, i, j/(j-2)!*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Jul 09 2022

a(n) ~ exp((n-r^3)/(2+r)-n) * n^(n+1/2) / (r^n * sqrt((2*r^3*(3+r) + n*(1+r)*(4+r))/(2+r))), where r is the root of the equation r^2*((2+r) * exp(r) - 2) = n.
(a(n)/n!)^(1/n) ~ exp(1/(3*LambertW(n^(1/3)/3))) / (3*LambertW(n^(1/3)/3)).
From Seiichi Manyama, Jul 09 2022: (Start)
a(n) = n! * Sum_{k=0..floor(n/3)} Stirling2(n-2*k,k)/(n-2*k)!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=3..n} k/(k-2)! * a(n-k)/(n-k)!. (End)

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