A240989 -id:A240989 - cp's OEIS frontend

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

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

A292892 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^k * (exp(x) - 1)).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 0, 2, 5, 1, 0, 0, 3, 15, 1, 0, 0, 6, 16, 52, 1, 0, 0, 0, 12, 65, 203, 1, 0, 0, 0, 24, 20, 336, 877, 1, 0, 0, 0, 0, 60, 390, 1897, 4140, 1, 0, 0, 0, 0, 120, 120, 2562, 11824, 21147, 1, 0, 0, 0, 0, 0, 360, 210, 11816, 80145, 115975, 1, 0, 0, 0, 0, 0, 720, 840, 20496, 105912, 586000, 678570

Offset: 0

Seiichi Manyama, Sep 26 2017

			Square array begins:
   1,  1,  1,  1,   1, ...
   1,  0,  0,  0,   0, ...
   2,  2,  0,  0,   0, ...
   5,  3,  6,  0,   0, ...
  15, 16, 12, 24,   0, ...
  52, 65, 20, 60, 120, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..3 give A000110, A052506, A240989, A292891.
Rows n=0..1 give A000012, A000007.
Main diagonal gives A000007.
Cf. A292894, A355607.

T(n, k) = n!*sum(j=0, n\(k+1), stirling(n-k*j, j, 2)/(n-k*j)!); \\ Seiichi Manyama, Jul 09 2022

From Seiichi Manyama, Jul 09 2022: (Start)
T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} Stirling2(n-k*j,j)/(n-k*j)!.
T(0,k) = 1 and T(n,k) = (n-1)! * Sum_{j=k+1..n} j/(j-k)! * T(n-j,k)/(n-j)! for n > 0. (End)

A358013 Expansion of e.g.f. 1/(1 - x^2 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 6, 12, 20, 750, 5082, 23576, 453672, 5755770, 50894030, 841270452, 14694142476, 201442729670, 3552604015170, 73814245552560, 1369932831933392, 27860865121662066, 655240785723048726, 15052226249248287500, 357713461766745539700, 9416426612423343023742

Offset: 0

Seiichi Manyama, Oct 24 2022

nonn

Cf. A052848, A358014.

Cf. A240989, A351503, A353998.

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

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

a(n) = n!*sum(k=0, n\3, k!*stirling(n-2*k, k, 2)/(n-2*k)!);

a(0) = 1; a(n) = n! * Sum_{k=3..n} 1/(k-2)! * a(n-k)/(n-k)!.

a(n) = n! * Sum_{k=0..floor(n/3)} k! * Stirling2(n-2*k,k)/(n-2*k)!.

A007121 Expansion of e.g.f. ( (1+x)^x )^x.

Original entry on oeis.org

1, 0, 0, 6, -12, 40, 180, -1512, 11760, -38880, 20160, 2106720, -22381920, 173197440, -703999296, -1737489600, 86030380800, -1149696737280, 11455162974720, -89560399541760, 636617260339200, -6318191386644480, 139398889956480000, -3797936822885990400

Offset: 0

Simon Plouffe

sign

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

Seiichi Manyama, Table of n, a(n) for n = 0..452

Cf. A240989.

A007121 := proc(n)
        n!*coeftayl( (1+x)^(x^2),x=0,n) ;
end proc:
seq(A007121(n),n=0..40) ; # R. J. Mathar, Dec 15 2011

With[{nn=30},CoefficientList[Series[((1+x)^x)^x,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 24 2014 *)

a(n):=sum(stirling1(n-2*k, k)/(n-2*k)!, k, 0, n/3); /* Vladimir Kruchinin, Dec 13 2011 */

a(n) = n!*sum(k=0, n\3, stirling(n-2*k, k, 1)/(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, (-1)^j*j/(j-2)*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Jul 09 2022

a(n) = n!*Sum_{k=0..floor(n/3)} Stirling1(n-2*k,k)/(n-2*k)!. - Vladimir Kruchinin, Dec 13 2011

a(0) = 1; a(n) = -(n-1)! * Sum_{k=3..n} (-1)^k * k/(k-2) * a(n-k)/(n-k)!. - Seiichi Manyama, Jul 09 2022

Signs added by R. J. Mathar, Vladimir Kruchinin, Dec 15 2011

A292951 Expansion of e.g.f. exp(x^2 * (1 - exp(x))).

Original entry on oeis.org

1, 0, 0, -6, -12, -20, 330, 2478, 11704, -15192, -751050, -7817150, -38408172, 151402524, 5793891922, 69046056870, 393083614320, -2517944476592, -98819987200146, -1384209703077750, -9376308260215220, 67288368700200900, 3186749671049174538

Offset: 0

Seiichi Manyama, Sep 27 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..200

Column k=2 of A292894.

Cf. A240989.

With[{nn=30},CoefficientList[Series[Exp[x^2 (1-Exp[x])],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jul 16 2021 *)

x='x+O('x^66); Vec(serlaplace(exp(x^2*(1-exp(x)))))

a(n) = n!*sum(k=0, n\3, (-1)^k*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

From Seiichi Manyama, Jul 09 2022: (Start)
a(n) = n! * Sum_{k=0..floor(n/3)} (-1)^k * 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)

A356949 E.g.f. satisfies log(A(x)) = x^2 * (exp(x) - 1) * A(x).

Original entry on oeis.org

1, 0, 0, 6, 12, 20, 1110, 7602, 35336, 1103832, 14984010, 134552990, 3457329612, 70828191876, 1017237973934, 25648737955050, 676111332667920, 13760810592066992, 373071111301807506, 11594147432172228918, 307097278689726728660, 9330499711181779575900

Offset: 0

Seiichi Manyama, Sep 06 2022

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A052880, A355843, A356950.

Cf. A000272, A240989, A355287, A356797, A356951.

nmax = 21; A[_] = 1;
Do[A[x_] = Exp[(-1 + Exp[x])*A[x]*x^2] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

a(n) = n!*sum(k=0, n\3, (k+1)^(k-1)*stirling(n-2*k, k, 2)/(n-2*k)!);

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(x^2*(exp(x)-1))^k/k!)))

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

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

a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)^(k-1) * Stirling2(n-2*k,k)/(n-2*k)!.
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (x^2 * (exp(x) - 1))^k / k!.
E.g.f.: A(x) = exp( -LambertW(x^2 * (1 - exp(x))) ).
E.g.f.: A(x) = LambertW(x^2 * (1 - exp(x)))/(x^2 * (1 - exp(x))).

A362892 Expansion of e.g.f. 1/(1 + LambertW(-x^2 * (exp(x) - 1))).

Original entry on oeis.org

1, 0, 0, 6, 12, 20, 1470, 10122, 47096, 1814472, 25119450, 226527950, 6732015972, 142901684796, 2071229736758, 57596022404130, 1589579741044080, 32832196825559312, 951335638952843826, 31043287459520549910, 838738470701197009820

Offset: 0

Seiichi Manyama, May 08 2023

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A282190, A362836.

Cf. A240989, A362891.

With[{nn=20},CoefficientList[Series[1/(1+LambertW[-x^2(Exp[x]-1)]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 14 2025 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(-x^2*(exp(x)-1)))))

a(n) = n! * Sum_{k=0..floor(n/3)} k^k * Stirling2(n-2*k,k)/(n-2*k)!.

a(n) ~ n^n / (sqrt((2+r)*exp(r) - 2) * r^(n+1) * exp(n + 1/2)), where r = 0.640353588740603511543638690178204955926349... is the root of the equation r^2*(exp(r+1) - exp(1)) = 1. - Vaclav Kotesovec, May 19 2025

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

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A292892 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^k * (exp(x) - 1)).

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A358013 Expansion of e.g.f. 1/(1 - x^2 * (exp(x) - 1)).

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A007121 Expansion of e.g.f. ( (1+x)^x )^x.

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A292951 Expansion of e.g.f. exp(x^2 * (1 - exp(x))).

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A356949 E.g.f. satisfies log(A(x)) = x^2 * (exp(x) - 1) * A(x).

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A362892 Expansion of e.g.f. 1/(1 + LambertW(-x^2 * (exp(x) - 1))).

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