A352410 -id:A352410 - cp's OEIS frontend

A052868 Expansion of e.g.f. LambertW(x/(-1+x))/x*(-1+x).

1, 1, 5, 40, 449, 6556, 118507, 2561518, 64540625, 1859206600, 60309007091, 2176222795594, 86488677518905, 3754431762036892, 176771908657345835, 8973513955735900246, 488586200931213192353, 28404347922603101834512

Offset: 0

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

Previous name was: A simple grammar.

G. C. Greubel, Table of n, a(n) for n = 0..369
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 839.

Cf. A352410, A370436.

spec := [S,{C=Sequence(Z,1 <= card),S=Set(B),B=Prod(C,S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

CoefficientList[Series[LambertW[x/(-1+x)]/x*(-1+x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 29 2013 *)
nmax = 20; A[] = 0; Do[A[x] = Product[Exp[x^k*A[x]], {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Mar 01 2024 *)

makelist(if n=0 then 1 else sum(n!/k!*binomial(n-1, k-1)*(k+1)^(k-1),k,0,n),n,0,17);  /* Bruno Berselli, May 25 2011 */

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

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1-x))))) \\ Seiichi Manyama, Mar 01 2023

E.g.f.: LambertW(x/(-1+x))/x*(-1+x).
a(n) = Sum_{k=0..n} n!/k!*binomial(n-1, k-1)*(k+1)^(k-1). - Vladeta Jovovic, Sep 17 2003
a(n) ~ sqrt((exp(1)+1)*exp(1))*n^(n-1)*(1+exp(-1))^n. - Vaclav Kotesovec, Sep 29 2013
E.g.f. A(x) satisfies  A(x) = exp( x*A(x)/(1-x) ) - Olivier Gérard, Dec 28 2013
E.g.f.: exp( -LambertW(-x/(1-x)) ). - Seiichi Manyama, Mar 01 2023

New name using e.g.f., Joerg Arndt, Sep 30 2013

A108919 Number of series-reduced labeled trees with n nodes.

Original entry on oeis.org

1, 0, 1, 1, 13, 51, 601, 4803, 63673, 775351, 12186061, 196158183, 3661759333, 72413918019, 1583407093633, 36916485570331, 929770285841137, 24904721121298671, 711342228666833173, 21502519995056598639, 687345492498807434461, 23135454269839313430715, 818568166383797223246601, 30357965273255025673685091

Offset: 1

Vladeta Jovovic, Jul 20 2005

"Series-reduced" means that if the tree is rooted at 1, then there is no node with just a single child.

Callan points out that A002792 is an incorrect version of this sequence. - Joerg Arndt, Jul 01 2014

Seiichi Manyama, Table of n, a(n) for n = 1..415
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], 2014.

The rooted version is A060356.

Cf. A000311, A000669, A001678, A292504, A316651, A316652, A318231, A318813, A330465, A330624, A352410.

f[n_] := Sum[(-1)^(n-k)*n!/k!*Binomial[n-1, k-1]*k^(k-1), {k, n}]/n; Table[ f[n], {n, 20}] (* Robert G. Wilson v, Jul 21 2005 *)

a(n) = { 1/n * sum(k=1, n, (-1)^(n-k) * binomial(n,k) * (n-1)!/(k-1)! * k^(k-1) ); } \\ Joerg Arndt, Aug 28 2014

a(n) = A060356(n)/n.
1 = Sum_{n>=0} a(n+1)*(exp(x)-x)^(-n-1)*x^n/n!.
E.g.f.: A(x) = Sum_{n>=0} a(n+1)*x^n/n! satisfies A(x) = exp(x*A(x))/(1+x). - Olivier Gérard, Dec 31 2013 (edited by Gus Wiseman, Dec 31 2019)
E.g.f.: -Integral (LambertW(-x/(1 + x))/x) dx. - Ilya Gutkovskiy, Jul 01 2020

More terms from Robert G. Wilson v, Jul 21 2005
New name (from A002792) by Joerg Arndt, Aug 28 2014
Offset corrected by Gus Wiseman, Dec 31 2019

A352448 Expansion of e.g.f. LambertW( -2x/(1-x) ) / (-2x).

Original entry on oeis.org

1, 3, 22, 278, 5128, 125592, 3850000, 142013328, 6129705088, 303238991744, 16920975718144, 1051612647426816, 72045481821580288, 5394849460316820480, 438392509692455286784, 38424395486908104071168, 3613476161122656804438016

Offset: 0

Paul D. Hanna, Mar 16 2022

nonn

An interesting property of this e.g.f. A(x) is that the sum of coefficients of x^k, k=0..n, in 1/A(x)^n equals zero, for n > 1.

			E.g.f.: A(x) = 1 + 3*x + 22*x^2/2! + 278*x^3/3! + 5128*x^4/4! + 125592*x^5/5! + 3850000*x^6/6! + 142013328*x^7/7! + ...
such that A(x) = exp( 2*x*A(x) ) / (1-x), where
exp( 2*x*A(x) ) = 1 + 2*x + 16*x^2/2! + 212*x^3/3! + 4016*x^4/4! + 99952*x^5/5! + 3096448*x^6/6! + 115063328*x^7/7! + ...
Related table.
Another interesting property of the e.g.f. A(x) is illustrated here.
The table of coefficients of x^k/k! in 1/A(x)^n begins:
n=1: [1,  -3,  -4,   -44,  -736,  -16832, -491168, ...];
n=2: [1,  -6,  10,   -16,  -320,   -8064, -249344, ...];
n=3: [1,  -9,  42,   -78,   -48,   -1776,  -66528, ...];
n=4: [1, -12,  92,  -392,   728,    -128,   -8960, ...];
n=5: [1, -15, 160, -1120,  4600,   -8520,    -320, ...];
n=6: [1, -18, 246, -2424, 16104,  -64752,  119952, ...];
...
from which we can illustrate that the partial sum of coefficients of x^k, k=0..n, in 1/A(x)^n equals zero, for n > 1, as follows:
n=1:-2 = 1 +  -3;
n=2: 0 = 1 +  -6 +  10/2!;
n=3: 0 = 1 +  -9 +  42/2! +   -78/3!;
n=4: 0 = 1 + -12 +  92/2! +  -392/3! +   728/4!;
n=5: 0 = 1 + -15 + 160/2! + -1120/3! +  4600/4! +   -8520/5!;
n=6: 0 = 1 + -18 + 246/2! + -2424/3! + 16104/4! +  -64752/5! +  119952/6!;
...

Seiichi Manyama, Table of n, a(n) for n = 0..344

Cf. A352410, A352411, A352412.

Cf. A361068.

terms = 17; A[] = 0; Do[A[x] = Exp[2x*A[x]]/(1-x) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Mar 24 2025 *)

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

my(x='x+O('x^30)); Vec(serlaplace(lambertw(-2*x/(1-x))/(-2*x))) \\ Michel Marcus, Mar 17 2022

a(n) = n!*sum(k=0, n, 2^k*(k+1)^(k-1)*binomial(n, k)/k!); \\ Seiichi Manyama, Mar 03 2023

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:
(1) A(x) = LambertW( -2*x/(1-x) ) / (-2*x).
(2) A(x) = exp( 2*x*A(x) ) / (1-x).
(3) A(x) = log( (1-x) * A(x) ) / (2*x).
(4) A( x/(exp(2*x) + x) ) = exp(2*x) + x.
(5) A(x) = (1/x) * Series_Reversion( x/(exp(2*x) + x) ).
(6) Sum_{k=0..n} [x^k] 1/A(x)^n = 0, for n > 1.
(7) [x^(n+1)/(n+1)!] 1/A(x)^n = -2^(n+1) * n for n >= (-1).
a(n) ~ (1 + 2*exp(1))^(n + 3/2) * n^(n-1) / (2^(3/2) * exp(n + 1/2)). - Vaclav Kotesovec, Mar 18 2022
a(n) = n! * Sum_{k=0..n} 2^k * (k+1)^(k-1) * binomial(n,k)/k!. - Seiichi Manyama, Mar 03 2023

A377826 E.g.f. satisfies A(x) = (1 + x) * exp(x * A(x)).

Original entry on oeis.org

1, 2, 7, 49, 489, 6521, 108643, 2178107, 51084337, 1373054833, 41624314371, 1405311853595, 52299954524953, 2127347522554073, 93902399411048803, 4470613587492385051, 228362858274694209249, 12458393118650371672673, 722983769486947261178371

Offset: 0

Seiichi Manyama, Nov 09 2024

nonn

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

Cf. A377827, A377828.

Cf. A352410, A362771.

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

E.g.f.: (1+x) * exp( -LambertW(-x*(1+x)) ).
E.g.f.: -LambertW(-x*(1+x))/x.
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(k+1,n-k)/k!.
a(n) ~ sqrt(-2*sqrt(1 + 4*exp(-1)) + 2 + 8*exp(-1)) * 2^n * n^(n-1) / ((-1 + sqrt(1 + 4*exp(-1)))^(n+1) * exp(n - 1/2)). - Vaclav Kotesovec, Nov 09 2024

A361182 E.g.f. satisfies A(x) = exp( 3xA(x) ) / (1-x).

Original entry on oeis.org

1, 4, 41, 735, 19293, 672573, 29342241, 1540097541, 94579646553, 6656561754345, 528414534842949, 46716837535074897, 4552821617337191637, 484953672676323320109, 56056228305888242732841, 6988787950179969557086797, 934866118278080385555647025

Offset: 0

Seiichi Manyama, Mar 03 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..329
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A352410, A352448.

Cf. A361066.

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[3*x*A[x]]/(1 - x) + 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*(k+1)^(k-1)*binomial(n, k)/k!);

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

a(n) = n! * Sum_{k=0..n} 3^k * (k+1)^(k-1) * binomial(n,k)/k!.
E.g.f.: LambertW( -3*x/(1-x) ) / (-3*x).
a(n) ~ (1 + 3*exp(1))^(n + 3/2) * n^(n-1) / (3^(3/2) * exp(n + 1/2)). - Vaclav Kotesovec, Mar 03 2023

A377811 E.g.f. satisfies A(x) = exp(x * A(x)) / (1 - x)^3.

Original entry on oeis.org

1, 4, 27, 283, 4217, 82971, 2041855, 60475885, 2096566449, 83324680435, 3736041351311, 186594364199277, 10274269171279657, 618386703880855339, 40393224245061185919, 2846030947359659421901, 215160957844217080056161, 17373449685399138641312739, 1492298627191467511376377999

Offset: 0

Seiichi Manyama, Nov 08 2024

nonn

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

Cf. A352410, A377810.

Cf. A082030, A367789, A377743.

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

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

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

A360601 E.g.f. satisfies A(x) = exp(x*A(x)^2) / (1-x).

Original entry on oeis.org

1, 2, 13, 166, 3265, 87306, 2957509, 121400350, 5857287937, 324884241874, 20370279663901, 1424790170536470, 109990236302275201, 9289460282062082266, 852049115732672006101, 84345608594930495005966, 8962937531710834906989313, 1017655033307013508626619554

Offset: 0

Seiichi Manyama, Mar 05 2023

nonn

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

Cf. A352410, A360609.
Cf. A052750, A352448.
Cf. A370875.

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

E.g.f.: sqrt(LambertW( -2*x/(1-x)^2 ) / (-2*x)).
a(n) ~ sqrt(1 + 2*exp(-1) - sqrt(1 + 2*exp(-1))) * n^(n-1) / (2 * (sqrt(1 + 2*exp(-1)) - 1)^(3/2) * exp(2*n + 1/2) * (1 + exp(-1) - sqrt(1 + 2*exp(-1)))^n). - Vaclav Kotesovec, Mar 06 2023
a(n) = n! * Sum_{k=0..n} (2*k+1)^(k-1) * binomial(n+k,n-k)/k!. - Seiichi Manyama, Mar 09 2024

A360609 E.g.f. satisfies A(x) = exp(x*A(x)^3) / (1-x).

Original entry on oeis.org

1, 2, 17, 313, 9053, 357941, 17975605, 1095604133, 78570635225, 6482415935449, 604889610870881, 62989604872166897, 7241672622495518773, 911048848278644776949, 124497704904842673086285, 18364053909500922198147421, 2908158473059042016441887025

Offset: 0

Seiichi Manyama, Mar 05 2023

nonn

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

Cf. A352410, A360601.
Cf. A052752, A361182.
Cf. A370876.

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

E.g.f.: (LambertW( -3*x/(1-x)^3 ) / (-3*x))^(1/3).
a(n) ~ 3^(-5/6) * (2^(4/3) + 2*(3 + sqrt(4*exp(1) + 9))^(1/3) * exp(-2/3) - 2^(2/3) * (3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3))^(1/6) * 2^(1/3) * (3 + sqrt(4*exp(1) + 9))^(4/9) * sqrt(4 - 2^(4/3) * (3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3) + 3*2^(2/3) * exp(-2/3) * (3 + sqrt(4*exp(1) + 9))^(1/3)) * n^(n-1) * (12 + 4*sqrt(4*exp(1) + 9))^(n/3) / (exp(7/18 + 5*n/3) * (2 - 2^(1/3) * (3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3) + exp(-2/3) * (12 + 4*sqrt(4*exp(1) + 9))^(1/3))^n * ((3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3) - 2^(2/3))^(3/2) * sqrt(2^(1/3) * (3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3) - 2)). - Vaclav Kotesovec, Mar 06 2023
a(n) = n! * Sum_{k=0..n} (3*k+1)^(k-1) * binomial(n+2*k,n-k)/k!. - Seiichi Manyama, Mar 09 2024

A377810 E.g.f. satisfies A(x) = exp(x * A(x)) / (1 - x)^2.

Original entry on oeis.org

1, 3, 17, 154, 1993, 34066, 728209, 18733926, 564117425, 19473863986, 758421401401, 32901791851006, 1573602042306265, 82267318018246986, 4667656830688700801, 285662368622361581206, 18758565855176593500385, 1315663025587514658845026, 98160436697525045768511721

Offset: 0

Seiichi Manyama, Nov 08 2024

nonn

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

Cf. A352410, A377811.

Cf. A362775, A377742.

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

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

E.g.f.: exp( -LambertW(-x/(1-x)^2) )/(1-x)^2.
E.g.f.: -LambertW(-x/(1-x)^2)/x.
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(n+k+1,n-k)/k!.
a(n) ~ 2^(n + 3/2) * sqrt(1 + 4*exp(-1) - sqrt(1 + 4*exp(-1))) * n^(n-1) / ((-1 + sqrt(1 + 4*exp(-1)))^(3/2) * (1 + 2*exp(-1) - sqrt(1 + 4*exp(-1)))^(n + 1/2) * exp(2*n+1)). - Vaclav Kotesovec, Nov 11 2024

A352411 E.g.f.: x / LambertW( x/(1-x) ).

Original entry on oeis.org

1, 0, -1, 1, -7, 31, -281, 2381, -28015, 346879, -5149009, 82769149, -1499707991, 29444151023, -632715633577, 14631547277101, -364321853163871, 9686058045625471, -274387229080161569, 8241211775883617405, -261766195805536280839, 8763341168691985628719

Offset: 0

Paul D. Hanna, Mar 15 2022

sign

An interesting property of this e.g.f. A(x) is that the sum of coefficients of x^k, k=0..n, in A(x)^n equals zero, for n > 1.

			E.g.f.: A(x) = 1 + 0*x - x^2/2! + x^3/3! - 7*x^4/4! + 31*x^5/5! - 281*x^6/6! + 2381*x^7/7! - 28015*x^8/8! + ...
such that A(x) = (1-x) * exp(x/A(x)), where
exp(x/A(x)) = 1 + x + x^2/2! + 4*x^3/3! + 9*x^4/4! + 76*x^5/5! + 175*x^6/6! + 3606*x^7/7! + 833*x^8/8! + ...
Related series.
The e.g.f. A(x) satisfies A( x/(exp(-x) + x) ) = 1/(exp(-x) + x), where
1/(exp(-x) + x) = 1 - x^2/2! + x^3/3! + 5*x^4/4! - 19*x^5/5! - 41*x^6/6! + 519*x^7/7! - 183*x^8/8! + ...
Related table.
Another defining property of the e.g.f. A(x) is illustrated here.
The table of coefficients of x^k/k! in A(x)^n begins:
n=1: [1, 0, -1, 1,  -7,   31, -281, 2381, -28015, ...];
n=2: [1, 0, -2, 2,  -8,   42, -332, 2970, -33392, ...];
n=3: [1, 0, -3, 3,  -3,   33, -243, 2397, -26631, ...];
n=4: [1, 0, -4, 4,   8,    4, -104, 1292, -15712, ...];
n=5: [1, 0, -5, 5,  25,  -45,   -5,  285,  -6095, ...];
n=6: [1, 0, -6, 6,  48, -114,  -36,    6,   -720, ...];
n=7: [1, 0, -7, 7,  77, -203, -287, 1085,     -7, ...];
n=8: [1, 0, -8, 8, 112, -312, -848, 4152,  -1856, 8, ...];
...
from which we can illustrate that the partial sum of coefficients of x^k, k=0..n, in A(x)^n equals zero, for n > 1, as follows:
n=1: 1 = 1 + 0;
n=2: 0 = 1 + 0 + -2/2!;
n=3: 0 = 1 + 0 + -3/2! + 3/3!;
n=4: 0 = 1 + 0 + -4/2! + 4/3! +   8/4!;
n=5: 0 = 1 + 0 + -5/2! + 5/3! +  25/4! +  -45/5!;
n=6: 0 = 1 + 0 + -6/2! + 6/3! +  48/4! + -114/5! +  -36/6!;
n=7: 0 = 1 + 0 + -7/2! + 7/3! +  77/4! + -203/5! + -287/6! + 1085/7!;
n=8: 0 = 1 + 0 + -8/2! + 8/3! + 112/4! + -312/5! + -848/6! + 4152/7! + -1856/8!;
...

Cf. A352410, A352412, A352448.

{a(n) = n!*polcoeff( x/serreverse( x/(exp(-x  +x^2*O(x^n)) + x) ),n)}
for(n=0,30,print1(a(n),", "))

my(x='x+O('x^30)); Vec(serlaplace(x/lambertw(x/(1-x)))) \\ Michel Marcus, Mar 17 2022

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:
(1) A(x) = x / LambertW( x/(1-x) ).
(2) A(x) = (1-x) * exp( x/A(x) ).
(3) A(x) = x / log( A(x)/(1-x) ).
(4) A( x/(exp(-x) + x) ) = 1/(exp(-x) + x).
(5) A(x) = x / Series_Reversion( x/(exp(-x) + x) ).
(6) Sum_{k=0..n} [x^k] A(x)^n = 0, for n > 1.
(7) [x^(n+1)/(n+1)!] A(x)^n = (-1)^n * n for n >= (-1).
a(n) ~ (-1)^(n+1) * exp(-1) * (1 - exp(-1))^(n - 1/2) * n^(n-1). - Vaclav Kotesovec, Mar 15 2022

cp's OEIS Frontend

A052868 Expansion of e.g.f. LambertW(x/(-1+x))/x*(-1+x).

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A108919 Number of series-reduced labeled trees with n nodes.

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A352448 Expansion of e.g.f. LambertW( -2*x/(1-x) ) / (-2*x).

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A377826 E.g.f. satisfies A(x) = (1 + x) * exp(x * A(x)).

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A361182 E.g.f. satisfies A(x) = exp( 3*x*A(x) ) / (1-x).

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A377811 E.g.f. satisfies A(x) = exp(x * A(x)) / (1 - x)^3.

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A360601 E.g.f. satisfies A(x) = exp(x*A(x)^2) / (1-x).

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A352448 Expansion of e.g.f. LambertW( -2x/(1-x) ) / (-2x).

A361182 E.g.f. satisfies A(x) = exp( 3xA(x) ) / (1-x).