A048802 -id:A048802 - cp's OEIS frontend

A052807 Expansion of e.g.f. -LambertW(log(1-x)).

0, 1, 3, 17, 146, 1704, 25284, 456224, 9702776, 237711888, 6593032560, 204212077992, 6986942528400, 261700394006232, 10650713784774504, 468007296229553880, 22083086552247101184, 1113646609708909274880

Offset: 0

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

Previous name was: A simple grammar.

E.g.f. of A052813 equals exp(A(x)) = -A(x)/log(1-x). a(n) = n!*Sum_{k=0..n-1} A052813(k)/k!/(n-k). - Paul D. Hanna, Jul 19 2006

			E.g.f.: A(x) = x + 3*x^2/2! + 17*x^3/3! + 146*x^4/4! +...
A(x)/exp(A(x)) = -log(1-x) = x + 1/2*x^2 + 1/3*x^3 + 1/4*x^4 +...

G. C. Greubel, Table of n, a(n) for n = 0..375
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 769

Cf. A006963, A048802, A052813 (exp(A(x))), A277489.

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

max = 17; se = Series[-ProductLog[-Log[-(-1 + x)^(-1)]] , {x, 0, max}]; Join[{0}, (CoefficientList[se, x] // DeleteCases[#, 0] &) * Range[max]!] (* Jean-François Alcover, Jun 24 2013 *)
CoefficientList[Series[-LambertW[-Log[-1/(-1 + x)]], {x,0,50}], x]*
Range[0,50]! (* G. C. Greubel, Jun 18 2017 *)

{a(n)=local(A=1+x);for(i=1,n,A=1/(1-x+x*O(x^n))^A);n!*polcoeff(log(A),n)} \\ Paul D. Hanna, Jul 19 2006

x = 'x + O('x^30); concat(0, Vec(serlaplace(-lambertw(log(1-x))))) \\ Michel Marcus, Jun 19 2017

a(n) = Sum_{k=1..n} |Stirling1(n, k)|*k^(k-1). - Vladeta Jovovic, Sep 17 2003
E.g.f. satisfies: A(x) = 1/(1-x)^A(x). - Paul D. Hanna, Jul 19 2006
a(n) ~ n^(n-1)*exp((exp(-1)-1)*n+1/2) / (exp(exp(-1))-1)^(n-1/2). - Vaclav Kotesovec, Jul 09 2013
E.g.f.: Series_Reversion( 1 - exp(-x*exp(-x)) ). - Seiichi Manyama, Sep 08 2024

New name using e.g.f. by Vaclav Kotesovec, Oct 18 2013

A216857 Number of connected functions from {1,2,...,n} into a subset of {1,2,...,n} that have a fixed point summed over all subsets.

Original entry on oeis.org

0, 1, 4, 24, 224, 2880, 47232, 942592, 22171648, 600698880, 18422374400, 630897721344, 23864653578240, 988197253808128, 44460603225407488, 2159714024218951680, 112652924603290615808, 6280048587936003784704, 372616014329572403183616, 23445082059018189741752320, 1559275240299007139066675200

Offset: 0

Geoffrey Critzer, Sep 17 2012

nonn

Essentially the same as A038049.
Also the number of rooted trees whose nodes are labeled with the blocks of a set partition of {1,2,...,n} each having a distinguished element. (See A000248.)
The bijection is straightforward. The trees correspond to functional digraphs mapping the distinguished elements towards the root. All the elements within each block are mapped to the distinguished element of that block. The distinguished element in the root node is the fixed point.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.

Cf. A048802, A072034.

With[{nmax = 20}, CoefficientList[Series[-LambertW[-x*Exp[x]], {x, 0, nmax}], x]*Range[0, nmax]!] (* modified by G. C. Greubel, Nov 15 2017 *)

for(n=0,30, print1(sum(k=1,n, binomial(n,k)*k^(n-1)), ", ")) \\ G. C. Greubel, Nov 15 2017

my(x='x+O('x^50)); concat([0], Vec(serlaplace(-lambertw(-x*exp(x))))) \\ G. C. Greubel, Nov 15 2017

E.g.f.: T(x*exp(x)) where T(x) is the e.g.f. for A000169.
a(n) = Sum_{k=1..n} binomial(n,k)*k^(n-1).
a(n) ~ sqrt(1+LambertW(exp(-1))) * n^(n-1) / (exp(n)*LambertW(exp(-1))^n). - Vaclav Kotesovec, Jul 09 2013
O.g.f.: Sum_{n>=0} n^(n-1)* x^n / (1 - n*x)^(n+1). - Paul D. Hanna, May 22 2018
E.g.f.: the compositional inverse of A(x) is -A(-x). - Alexander Burstein, Aug 11 2018

A036249 Number of rooted trees of nonempty sets with n points. (Each node is a set of 1 or more points.)

Original entry on oeis.org

0, 1, 2, 5, 13, 37, 108, 332, 1042, 3360, 11019, 36722, 123875, 422449, 1453553, 5040816, 17599468, 61814275, 218252584, 774226549, 2758043727, 9862357697, 35387662266, 127374191687, 459783039109, 1664042970924, 6037070913558, 21951214425140, 79981665585029

Offset: 0

Christian G. Bower, Nov 15 1998

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1717
Håvard Berland, Brynjulf Owren and Bård Skaflestad, B-series and order conditions for exponential integrators, 2004. See p. 6.
F. Chapoton, F. Hivert, and J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.
F. Chapoton, F. Hivert, and J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, Journal of Algebra, 465 (2016), 322-355.
Timothy Y. Chow and Mark G. Tiefenbruck, The Latin Tableau Conjecture, 2024. See p. 11.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 768
Index entries for sequences related to rooted trees

Essentially the same as A029856. Cf. A048802. Row sums of A303911.

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*
      add(d*a(d), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+b(n-1)) end:
seq(a(n), n=0..35);  # Alois P. Heinz, Jun 13 2018

max = 27; A[] = 1; Do[A[x] = x*Exp[Sum[(A[x^k] + x^k)/k + O[x]^n, {k, 1, n}]] // Normal, {n, 1, max}]; CoefficientList[A[x] + O[x]^max, x] (* Jean-François Alcover, May 25 2018 *)

{a(n)=local(A=x+x*O(x^n));for(i=1,n, A=x*exp(sum(m=1,n,(subst(A,x,x^m)+x^m)/m)));polcoeff(A,n,x)} \\ Paul D. Hanna, Oct 19 2005

G.f. satisfies: A(x) = x*exp( Sum_{n>=1} (A(x^n) + x^n)/n ). - Paul D. Hanna, Oct 19 2005
If b(n) is the Euler transform of a(n), A052855, then a(n+1) = a(n) + b(n). - Franklin T. Adams-Watters, Mar 09 2006
G.f.: (x/(1 - x)) * Product_{n>=1} 1/(1 - x^n)^a(n). - Ilya Gutkovskiy, Jun 28 2021

A282190 E.g.f.: 1/(1 + LambertW(1-exp(x))), where LambertW() is the Lambert W-function.

Original entry on oeis.org

1, 1, 5, 40, 447, 6421, 112726, 2338799, 55990213, 1519122598, 46066158817, 1543974969769, 56677405835276, 2261488166321697, 97455090037460785, 4510770674565054000, 223183550978156866507, 11755122645815049275521, 656670295411196201190366, 38779502115371642484125915, 2413908564514961126280655257

Offset: 0

Ilya Gutkovskiy, Feb 08 2017

Stirling transform of A000312.

			E.g.f.: A(x) = 1 + x/1! + 5*x^2/2! + 40*x^3/3! + 447*x^4/4! + 6421*x^5/5! + 112726*x^6/6! + ...

G. C. Greubel, Table of n, a(n) for n = 0..375
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Eric Weisstein's MathWorld, Stirling Transform

Cf. A000312, A000670, A038052, A048802, A052880, A308490, A308491.

b:= proc(n, m) option remember;
     `if`(n=0, m^m, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 03 2021

Range[0, 20]! CoefficientList[Series[1/(1 + ProductLog[1 - Exp[x]]), {x, 0, 20}], x]
Join[{1}, Table[Sum[StirlingS2[n, k] k^k, {k, 1, n}], {n, 1, 20}]]

x='x+O('x^50); Vec(serlaplace(1/(1 + lambertw(1-exp(x))))) \\ G. C. Greubel, Nov 12 2017

a(0) = 1, a(n) = Sum_{k=1..n} Stirling2(n,k)*k^k.

a(n) ~ n^n / (sqrt(1+exp(1)) * (log(1+exp(-1)))^(n+1/2) * exp(n)). - Vaclav Kotesovec, Feb 17 2017

A038052 Number of labeled trees of nonempty sets with n points. (Each node is a set of 1 or more points.)

Original entry on oeis.org

1, 1, 2, 7, 42, 376, 4513, 68090, 1238968, 26416729, 646140364, 17837852044, 548713088399, 18612963873492, 690271321314292, 27785827303491579, 1206582732097720126, 56224025231569020724, 2798445211000659147033, 148178324442139816854902, 8317074395027724691495980

Offset: 0

Christian G. Bower, Jan 04 1999

Alois P. Heinz, Table of n, a(n) for n = 0..379 (first 101 terms from T. D. Noe)
Index entries for sequences related to trees

Cf. A036250, A048802.

b:= proc(n, m) option remember; `if`(n=0,
      m^max(0, m-2), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..21);  # Alois P. Heinz, Jul 16 2022

a[0] = 1; a[n_] := Sum[StirlingS2[n, k]*k^(k - 2), {k, 1, n}]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Sep 09 2013, after Vladeta Jovovic *)

E.g.f.: B(e^x-1) where B is e.g.f. of A000272.
a(n) = Sum_{k=1..n} Stirling2(n, k)*k^(k-2). - Vladeta Jovovic, Sep 20 2003
a(n) ~ (1+exp(1))^(3/2) * n^(n-2) / (exp(n) * (log(1+exp(-1)))^(n-3/2)). - Vaclav Kotesovec, Feb 17 2017

A058863 Number of connected labeled chordal graphs on n nodes with no induced path P_4; also the number of labeled trees with each vertex replaced by a clique.

Original entry on oeis.org

1, 1, 4, 23, 181, 1812, 22037, 315569, 5201602, 97009833, 2019669961, 46432870222, 1168383075471, 31939474693297, 942565598033196, 29866348653695203, 1011335905644178273, 36446897413531401020, 1392821757824071815641, 56259101478392975833333

Offset: 1

Robert Castelo, Jan 06 2001

nonn

A subclass of chordal-comparability graphs.

Jon E. Schoenfield, Table of n, a(n) for n = 1..100
R. Castelo and N. C. Wormald, Enumeration of P4-free chordal graphs
R. Castelo and N. C. Wormald, Enumeration of P4-Free chordal graphs, Graphs and Combinatorics, 19:467-474, 2003.
M. C. Golumbic, Trivially perfect graphs, Discr. Math. 24(1) (1978), 105-107.
Venkatesan Guruswami, Enumerative aspects of certain subclasses of perfect graphs, Discrete Math. 205 (1999), 97-117.
T. H. Ma and J. P. Spinrad, Cycle-free partial orders and chordal comparability graphs, Order, 1991, 8:49-61.
E. S. Wolk, A note on the comparability graph of a tree, Proc. Am. Math. Soc., 1965, 16:17-20.

Cf. A007134, A058864, A058865.

Cf. A048802.

S:= series(-LambertW(exp(-x)-1), x, 101):
seq(coeff(S,x,j)*j!, j=1..100); # Robert Israel, Nov 30 2015

a[n_] := Sum[(-1)^(n-k)*StirlingS2[n, k]*k^(k-1), {k, 1, n}];
Array[a, 20] (* Jean-François Alcover, Dec 17 2017, after Vladeta Jovovic *)

geta(n, va, vA) = {local(k); if (n==1, return(1)); if (n==2, return(1)); return(1 + sum(k=1, n-2, binomial(n,k)*(vA[n-k] - va[n-k])));}
getA(n, va, vA) = {local(k); if (n==1, return(1)); if (n==2, return(2)); return ((va[n] + sum(k=1, n-1, k*va[k]*binomial(n,k)*vA[n-k])/n));}
both(n) = {va = vector(n); vA = vector(n); for (i=1, n, va[i] = geta(i, va, vA); vA[i] = getA(i, va, vA);); print("va_A058863=", va); print("vA_A058864=", vA);}
\\ Michel Marcus, Apr 03 2013

A058863 and A058864 satisfy:
  1) c(n) = 1 + Sum_{k=1..n-2} binomial(n, k)*(t(n-k) - c(n-k))
  2) t(n) = c(n) + Sum_{k=1..n-1} k*c(k)*binomial(n, k)*t(n-k)/n
  where c(n) (A058863) is the number of connected graphs of this type and t(n) (A058864) is the total number of such graphs.
a(n) is asymptotic to sqrt(r*(e-1))/n*(n/(e*r))^n where r = 1 - log(e-1).
E.g.f.: -LambertW(exp(-x)-1). - Vladeta Jovovic, Nov 22 2002
a(n) = Sum_{k=0..n} Stirling2(n, k)*A060356(k). Also a(n) = Sum_{k=1..n} (-1)^(n-k)*Stirling2(n, k)*k^(k-1). - Vladeta Jovovic, Sep 17 2003

Formulae edited and completed by Michel Marcus, Apr 07 2013

A357267 Expansion of e.g.f. -LambertW(x * (1 - exp(x))).

Original entry on oeis.org

0, 0, 2, 3, 28, 125, 1506, 12607, 186600, 2352681, 41839750, 705821171, 14818593516, 311784460429, 7603945309338, 190868446707135, 5328147004384336, 154893585657590609, 4884408906341245326, 161057122218190660555, 5671407469802947722900

Offset: 0

Seiichi Manyama, Sep 21 2022

nonn

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

Cf. A048802, A355843, A357265.

my(N=20, x='x+O('x^N)); concat([0, 0], Vec(serlaplace(-lambertw(x*(1-exp(x))))))

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

a(n) = n! * Sum_{k=1..floor(n/2)} k^(k-1) * Stirling2(n-k,k)/(n-k)!.

A038051 G.f.: B(x/(1-x)) where B is g.f. of A000169.

Original entry on oeis.org

1, 3, 14, 98, 944, 11642, 175108, 3108310, 63601168, 1473864722, 38152990484, 1091172974102, 34169139856024, 1162736848398010, 42723615842296540, 1685853467536076798, 71101435046807892512, 3191843270961299033762, 151956292916451992949028

Offset: 1

Christian G. Bower, Jan 04 1999

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A036249, A038052.

E.g.f. of A048802.

CoefficientList[Series[E^x*(-LambertW[-x]/(1+LambertW[-x])/x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Feb 17 2014 *)

E.g.f.: int(exp(x)*(-LambertW(-x)/(1+LambertW(-x))/x), x). a(n) = Sum_{k=0..n-1} binomial(n-1, k)*(k+1)^k. - Vladeta Jovovic, Apr 12 2003

a(n) ~ n^(n-1) * exp(exp(-1)). - Vaclav Kotesovec, Feb 17 2014

Corrected by Christian G. Bower, Mar 15 1999

A357335 E.g.f. satisfies A(x) = (exp(x) - 1) * exp(2 * A(x)).

Original entry on oeis.org

0, 1, 5, 49, 757, 16081, 435477, 14345297, 556857973, 24894290257, 1259621627349, 71165987957329, 4440821632449077, 303338709537825105, 22512353926895739797, 1803812930088064925265, 155195078834104237961717, 14270228623788585753803089

Offset: 0

Seiichi Manyama, Sep 24 2022

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.
Index entries for reversions of series

Cf. A048802, A357336.
Cf. A349524, A356000.
Cf. A357347.

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(2*(1-exp(x)))/2)))

a(n) = sum(k=1, n, (2*k)^(k-1)*stirling(n, k, 2));

E.g.f.: -LambertW(2 * (1 - exp(x)))/2.
a(n) = Sum_{k=1..n} (2 * k)^(k-1) * Stirling2(n,k).
a(n) ~ sqrt(1 + 2*exp(1)) * n^(n-1) / (2 * exp(n) * log(1 + exp(-1)/2)^(n - 1/2)). - Vaclav Kotesovec, Nov 14 2022
E.g.f.: Series_Reversion( log(1 + x * exp(-2*x)) ). - Seiichi Manyama, Sep 09 2024

A357336 E.g.f. satisfies A(x) = (exp(x) - 1) * exp(3 * A(x)).

Original entry on oeis.org

0, 1, 7, 100, 2257, 70021, 2768740, 133164109, 7546722487, 492531820066, 36381833190223, 3000677194970137, 273342303933512362, 27256107730344331879, 2952882035628632383975, 345384835617231362018764, 43378466647737203462409829, 5822506028894124326533926193

Offset: 0

Seiichi Manyama, Sep 24 2022

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.
Index entries for reversions of series

Cf. A048802, A357335.

Cf. A349525, A356001.

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(3*(1-exp(x)))/3)))

a(n) = sum(k=1, n, (3*k)^(k-1)*stirling(n, k, 2));

E.g.f.: -LambertW(3 * (1 - exp(x)))/3.
a(n) = Sum_{k=1..n} (3 * k)^(k-1) * Stirling2(n,k).
a(n) ~ sqrt(1 + 3*exp(1)) * n^(n-1) / (3 * exp(n) * log(1 + exp(-1)/3)^(n - 1/2)). - Vaclav Kotesovec, Nov 14 2022
E.g.f.: Series_Reversion( log(1 + x * exp(-3*x)) ). - Seiichi Manyama, Sep 09 2024

cp's OEIS Frontend

A052807 Expansion of e.g.f. -LambertW(log(1-x)).

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A216857 Number of connected functions from {1,2,...,n} into a subset of {1,2,...,n} that have a fixed point summed over all subsets.

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A036249 Number of rooted trees of nonempty sets with n points. (Each node is a set of 1 or more points.)

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A282190 E.g.f.: 1/(1 + LambertW(1-exp(x))), where LambertW() is the Lambert W-function.

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A038052 Number of labeled trees of nonempty sets with n points. (Each node is a set of 1 or more points.)

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A058863 Number of connected labeled chordal graphs on n nodes with no induced path P_4; also the number of labeled trees with each vertex replaced by a clique.

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A357267 Expansion of e.g.f. -LambertW(x * (1 - exp(x))).

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