A052868 -id:A052868 - cp's OEIS frontend

A352410 Expansion of e.g.f. LambertW( -x/(1-x) ) / (-x).

1, 2, 9, 67, 717, 10141, 179353, 3816989, 95076537, 2714895433, 87457961421, 3138260371225, 124147801973605, 5368353187693757, 251928853285058433, 12752446755011776741, 692625349011401620209, 40178978855796929378065, 2479383850197948228950293

Offset: 0

Paul D. Hanna, Mar 15 2022

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 + 2*x + 9*x^2/2! + 67*x^3/3! + 717*x^4/4! + 10141*x^5/5! + 179353*x^6/6! + 3816989*x^7/7! + ...
such that A(x) = exp(x*A(x)) / (1-x), where
exp(x*A(x)) = 1 + x + 5*x^2/2! + 40*x^3/3! + 449*x^4/4! + 6556*x^5/5! + 118507*x^6/6! + ... + A052868(n)*x^n/n! + ...
which equals LambertW(-x/(1-x)) * (1-x)/(-x).
Related table.
Another defining 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,  -2,  -1,    -7,   -71,   -961, -16409, -339571, ...];
n=2: [1,  -4,   6,    -2,   -24,   -362,  -6644, -144538, ...];
n=3: [1,  -6,  21,   -33,    -3,    -63,  -1395,  -34275, ...];
n=4: [1,  -8,  44,  -148,   232,     -4,   -152,   -4876, ...];
n=5: [1, -10,  75,  -395,  1305,  -2045,     -5,    -355, ...];
n=6: [1, -12, 114,  -822,  4224, -13806,  21636,      -6, ...];
n=7: [1, -14, 161, -1477, 10381, -52507, 170401, -267043, -7, ...];
...
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:-1 = 1 +  -2;
n=2: 0 = 1 +  -4 +   6/2!;
n=3: 0 = 1 +  -6 +  21/2! +   -33/3!;
n=4: 0 = 1 +  -8 +  44/2! +  -148/3! +   232/4!;
n=5: 0 = 1 + -10 +  75/2! +  -395/3! +  1305/4! +  -2045/5!;
n=6: 0 = 1 + -12 + 114/2! +  -822/3! +  4224/4! + -13806/5! +  21636/6!;
n=7: 0 = 1 + -14 + 161/2! + -1477/3! + 10381/4! + -52507/5! + 170401/6! + -267043/7!;
...

Seiichi Manyama, Table of n, a(n) for n = 0..370

Cf. A352411, A352412, A352448, A052868.

Cf. A102743, A108919, A331726.

terms = 19; A[] = 0; Do[A[x] = Exp[x*A[x]]/(1-x) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Mar 24 2025 *)
With[{nn=20},CoefficientList[Series[LambertW[-x/(1-x)]/-x,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 24 2025 *)

{a(n) = n!*polcoeff( (1/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(lambertw(-x/(1-x))/(-x))) \\ Michel Marcus, Mar 17 2022

a(n) = n!*sum(k=0, n, (k+1)^(k-1)*binomial(n, k)/k!); \\ Seiichi Manyama, Sep 24 2022

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:
(1) A(x) = LambertW( -x/(1-x) ) / (-x).
(2) A(x) = exp( x*A(x) ) / (1-x).
(3) A(x) = log( (1-x) * A(x) ) / x.
(4) A( x/(exp(x) + x) ) = exp(x) + x.
(5) A(x) = (1/x) * Series_Reversion( x/(exp(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 = -n for n >= (-1).
a(n) ~ (1 + exp(1))^(n + 3/2) * n^(n-1) / exp(n + 1/2). - Vaclav Kotesovec, Mar 15 2022
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(n,k)/k!. - Seiichi Manyama, Sep 24 2022

A362771 E.g.f. satisfies A(x) = exp( x * (1+x) * A(x) ).

Original entry on oeis.org

1, 1, 5, 34, 353, 4756, 80107, 1617358, 38145473, 1029745576, 31326858611, 1060716408874, 39571357618465, 1612919873514028, 71321521181852411, 3400790769764598886, 173950205958460627073, 9501239617356541012432, 551961456374529522954595

Offset: 0

Seiichi Manyama, May 02 2023

nonn

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

Cf. A052868, A125500.

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x*(1 + x)*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

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

E.g.f.: exp( -LambertW(-x * (1+x)) ).
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(k,n-k)/k!.
a(n) ~ sqrt(2 + 8*exp(-1) - 2*sqrt(1 + 4*exp(-1))) * 2^(n-1) * n^(n-1) / ((sqrt(1 + 4*exp(-1)) - 1)^n * exp(n - 3/2)). - Vaclav Kotesovec, May 03 2023

A052873 E.g.f. A(x) satisfies A(x) = exp(xA(x)/(1 - xA(x))).

Original entry on oeis.org

1, 1, 5, 46, 629, 11496, 263857, 7301680, 236748969, 8806142080, 369714769181, 17296339048704, 892335712777885, 50333180563864576, 3081739132775658825, 203555129140352505856, 14428195498061848405073, 1092403962489972428144640, 87990832863810814525250869

Offset: 0

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

Previous name was: A simple grammar.

Seiichi Manyama, Table of n, a(n) for n = 0..357
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 844.
R. Lorentz, S. Tringali, C. H. Yan, Generalized Goncarov polynomials, arXiv preprint arXiv:1511.04039 [math.CO], 2015.

Cf. A052868, A088690, A161630.

Cf. A382036, A382037, A382038.

spec := [S,{C=Sequence(B,1 <= card),S=Set(C),B=Prod(Z,S)},labeled]:
seq(combstruct[count](spec,size=n), n=0..20);
# Alternatively:
a := n -> `if`(n=0,1, n!*hypergeom([1-n],[2],-n-1)):
seq(simplify(a(n)), n=0..16); # Peter Luschny, Apr 20 2016

Table[Sum[(n+1)^(k-1)*n!/k!*Binomial[n-1,k-1],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 08 2014 *)

{a(n)=if(n==0,1,sum(k=0,n,(n+1)^(k-1)*n!/k!*binomial(n-1,k-1)))} \\ Paul D. Hanna, Sep 08 2012

{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,(m+1)^(m-1)*x^m/m!/(1-x*A+x*O(x^n))^m));n!*polcoeff(A,n)} \\ Paul D. Hanna, Sep 08 2012

E.g.f.: exp(RootOf(exp(_Z)*x*_Z+exp(_Z)*x-_Z)).
1 = Sum_{n>=0} a(n)*exp((n+1)*x/(x-1))*x^n/n!. - Vladeta Jovovic, Jul 20 2005
a(n) = Sum_{k=0..n} (n+1)^(k-1)*n!/k!*binomial(n-1,k-1). - Vladeta Jovovic, Jul 02 2006
E.g.f. satisfies: A(x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n! / (1-x*A(x))^n. - Paul D. Hanna, Sep 08 2012
Equivalently:
E.g.f. satisfies: A(x) = exp( x*A(x)/(1 - x*A(x)) ). - Olivier Gérard, Dec 29 2013
a(n) ~ (sqrt(5)-1) * 2^(n-1/2) * n^(n-1) * exp((sqrt(5)-1 + (sqrt(5)-3)*n)/2) / (5^(1/4) * (3-sqrt(5))^(n+1/2)). - Vaclav Kotesovec, Jan 08 2014
a(n) = n!*hypergeom([1-n],[2],-n-1) for n >= 1. - Peter Luschny, Apr 20 2016
E.g.f.: exp( Series_Reversion( x*exp(-x)/(1+x) ) ). - Seiichi Manyama, Mar 15 2025

New name using e.g.f., Vaclav Kotesovec, Jan 08 2014

A361065 E.g.f. satisfies A(x) = exp( (x/(1-x)) * A(x)^2 ).

Original entry on oeis.org

1, 1, 7, 85, 1521, 36421, 1097743, 39968601, 1707558401, 83777885929, 4643185678551, 286930307457949, 19562851003118833, 1458832806486727725, 118121195050068075167, 10320576944751955718881, 967863775658734350214017, 96970880819175875321264209

Offset: 0

Seiichi Manyama, Mar 01 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..342
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A052868, A361066, A361067, A361068, A361069.

A361065 := proc(n)
    add((2*k+1)^(k-1)*binomial(n-1,n-k)/k!,k=0..n) ;
    %*n! ;
end proc:
seq(A361065(n),n=0..10) ; # R. J. Mathar, Mar 02 2023

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[(x/(1 - x))*A[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, (2*k+1)^(k-1)*binomial(n-1, n-k)/k!);

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

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

a(n) = n! * Sum_{k=0..n} (2*k+1)^(k-1) * binomial(n-1,n-k)/k!.
E.g.f.: exp( -LambertW(-2*x/(1-x))/2 ).
E.g.f.: sqrt( -(1-x)/(2*x) * LambertW(-2*x/(1-x)) ).
a(n) ~ (1 + 2*exp(1))^(n + 1/2) * n^(n-1) / (2^(3/2) * exp(n)). - Vaclav Kotesovec, Mar 02 2023

A361066 E.g.f. satisfies A(x) = exp( (x/(1-x)) * A(x)^3 ).

Original entry on oeis.org

1, 1, 9, 148, 3673, 123276, 5234599, 269262022, 16279709793, 1131627199816, 88926737901031, 7796168316687906, 754414052156289265, 79872584117422215484, 9184299004593618881655, 1139822558262829096519726, 151857077047173825979147969

Offset: 0

Seiichi Manyama, Mar 01 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..327
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A052868, A361065, A361067, A361068, A361069.

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

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

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

a(n) = n! * Sum_{k=0..n} (3*k+1)^(k-1) * binomial(n-1,n-k)/k!.
E.g.f.: exp( -LambertW(-3*x/(1-x))/3 ).
E.g.f.: ( -(1-x)/(3*x) * LambertW(-3*x/(1-x)) )^(1/3).
a(n) ~ (1 + 3*exp(1))^(n + 1/2) * n^(n-1) / (3^(3/2) * exp(n + 1/6)). - Vaclav Kotesovec, Mar 02 2023

A361068 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)^2) ).

Original entry on oeis.org

1, 1, -1, 13, -127, 2101, -41801, 1030177, -29820127, 995977801, -37660751569, 1590847310581, -74242656468575, 3793664894534269, -210656932372422745, 12630986901470435401, -813335155262348743231, 55977540398642247218449

Offset: 0

Seiichi Manyama, Mar 01 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..360
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A052868, A361065, A361066, A361067, A361069.

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x/((1 - x)*A[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, (-2*k+1)^(k-1)*binomial(n-1, n-k)/k!);

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

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

a(n) = n! * Sum_{k=0..n} (-2*k+1)^(k-1) * binomial(n-1,n-k)/k!.
E.g.f.: exp( LambertW(2*x/(1-x))/2 ).
E.g.f.: 1 / sqrt( (1-x)/(2*x) * LambertW(2*x/(1-x)) ).
a(n) ~ (-1)^(n+1) * 2^(-3/2) * exp(-1/2) * (2 - exp(-1))^(n + 1/2) * n^(n-1). - Vaclav Kotesovec, Apr 22 2024

A361069 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)^3) ).

Original entry on oeis.org

1, 1, -3, 40, -719, 18396, -598157, 23713726, -1108701519, 59735988424, -3644505746549, 248358786667674, -18697767289462967, 1541202721786228060, -138046868771541971373, 13351368704222195975206, -1386710317839048140282783, 153939247458296219191539984

Offset: 0

Seiichi Manyama, Mar 01 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..338
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A052868, A361065, A361066, A361067, A361068.

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

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

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

a(n) = n! * Sum_{k=0..n} (-3*k+1)^(k-1) * binomial(n-1,n-k)/k!.
E.g.f.: exp( LambertW(3*x/(1-x))/3 ).
E.g.f.: 1 / ( (1-x)/(3*x) * LambertW(3*x/(1-x)) )^(1/3).
a(n) ~ (-1)^(n+1) * 3^(-3/2) * exp(-1/3) * (3 - exp(-1))^(n + 1/2) * n^(n-1). - Vaclav Kotesovec, Apr 22 2024

A361067 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)) ).

Original entry on oeis.org

1, 1, 1, 4, 9, 76, 175, 3606, 833, 354376, -1605249, 65111410, -718371071, 20105327100, -351241054177, 9362931464446, -214514949732735, 6039303900168976, -165679758877120001, 5093296357218337386, -159900268661169533119, 5405435526807425433220

Offset: 0

Seiichi Manyama, Mar 01 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..413
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A052868, A361065, A361066, A361068, A361069.

nmax = 21; A[_] = 1;
Do[A[x_] = Exp[x/((1 - x)*A[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, (-k+1)^(k-1)*binomial(n-1, n-k)/k!);

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

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

a(n) = n! * Sum_{k=0..n} (-k+1)^(k-1) * binomial(n-1,n-k)/k!.
E.g.f.: exp( LambertW(x/(1-x)) ).
E.g.f.: 1 / ( (1-x)/x * LambertW(x/(1-x)) ).
a(n) ~ -(-1)^n * exp(-1) * (1 - exp(-1))^(n + 1/2) * n^(n-1). - Vaclav Kotesovec, Mar 02 2023

A361142 E.g.f. satisfies A(x) = exp( xA(x)^2/(1 - xA(x)) ).

Original entry on oeis.org

1, 1, 7, 91, 1773, 46401, 1529593, 60911103, 2845757449, 152663425633, 9250206248781, 624880915165959, 46569571425664477, 3795729136868379777, 335902071304953561073, 32074779600414913885231, 3287242849289861637185937, 359917016243351870997841473

Offset: 0

Seiichi Manyama, Mar 02 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..338

Cf. A000262, A361143.

Cf. A052868, A052873, A161630.

Table[n! * Sum[(n+k+1)^(k-1) * Binomial[n-1,n-k]/k!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 03 2023 *)

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

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

a(n) ~ s^2 * sqrt((2 - r*s)/(2 + r*s*(-2 + s*(2 - r*s)^2))) * n^(n-1) / (exp(n) * r^(n - 1/2)), where r = 0.14220768719194290600038416000340972911571484385125... and s = 1.549730657609106944767484487465870359529391502493... are roots of the system of equations exp(r*s^2/(1 - r*s)) = s, r*s^2*(2 - r*s) = (1 - r*s)^2. - Vaclav Kotesovec, Mar 03 2023

A367789 E.g.f. satisfies A(x) = exp( x/(1-x)^3 * A(x) ).

Original entry on oeis.org

1, 1, 9, 106, 1697, 35076, 893947, 27165706, 960298593, 38751082552, 1758831242291, 88726543365054, 4926355857050641, 298605321687360676, 19623211558172733435, 1389870724939251455506, 105556814502357807727553, 8557797733469700008170224

Offset: 0

Seiichi Manyama, Nov 30 2023

nonn

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

Cf. A052868, A362775, A367790.

Cf. A091695, A360100.

A367789 := proc(n)
    n!*add((k+1)^(k-1) * binomial(n+2*k-1,n-k)/k!,k=0..n) ;
end proc:
seq(A367789(n),n=0..70) ; # R. J. Mathar, Dec 04 2023

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

E.g.f.: exp( -LambertW(-x/(1-x)^3) ).

a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(n+2*k-1,n-k)/k!.

cp's OEIS Frontend

A352410 Expansion of e.g.f. LambertW( -x/(1-x) ) / (-x).

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A362771 E.g.f. satisfies A(x) = exp( x * (1+x) * A(x) ).

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A052873 E.g.f. A(x) satisfies A(x) = exp(x*A(x)/(1 - x*A(x))).

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A361065 E.g.f. satisfies A(x) = exp( (x/(1-x)) * A(x)^2 ).

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A361066 E.g.f. satisfies A(x) = exp( (x/(1-x)) * A(x)^3 ).

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A361068 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)^2) ).

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A361069 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)^3) ).

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A052873 E.g.f. A(x) satisfies A(x) = exp(xA(x)/(1 - xA(x))).

A361142 E.g.f. satisfies A(x) = exp( xA(x)^2/(1 - xA(x)) ).