A052871 -id:A052871 - cp's OEIS frontend

A060356 Expansion of e.g.f.: -LambertW(-x/(1+x)).

0, 1, 0, 3, 4, 65, 306, 4207, 38424, 573057, 7753510, 134046671, 2353898196, 47602871329, 1013794852266, 23751106404495, 590663769125296, 15806094859299329, 448284980183376078, 13515502344669830287

Offset: 0

Vladeta Jovovic, Apr 01 2001

Also the number of labeled lone-child-avoiding rooted trees with n nodes. A rooted tree is lone-child-avoiding if it has no unary branchings, meaning every non-leaf node covers at least two other nodes. The unlabeled version is A001678(n + 1). - Gus Wiseman, Jan 20 2020

			From _Gus Wiseman_, Dec 31 2019: (Start)
Non-isomorphic representatives of the a(7) = 4207 trees, written as root[branches], are:
  1[2,3[4,5[6,7]]]
  1[2,3[4,5,6,7]]
  1[2[3,4],5[6,7]]
  1[2,3,4[5,6,7]]
  1[2,3,4,5[6,7]]
  1[2,3,4,5,6,7]
(End)

Harry J. Smith, Table of n, a(n) for n = 0..100
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Cf. A052871, A060313.
Cf. A008297.
Column k=0 of A231602.
The unlabeled version is A001678(n + 1).
The case where the root is fixed is A108919.
Unlabeled rooted trees are counted by A000081.
Lone-child-avoiding rooted trees with labeled leaves are A000311.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Singleton-reduced rooted trees are counted by A330951.
Cf. A000669, A004111, A005121, A048816, A292504, A316651, A316652, A318231, A318813, A330465, A330624.

List([0..20],n->Sum([1..n],k->(-1)^(n-k)*Factorial(n)/Factorial(k) *Binomial(n-1,k-1)*k^(k-1))); # Muniru A Asiru, Feb 19 2018

seq(coeff(series( -LambertW(-x/(1+x)), x, n+1), x, n)*n!, n = 0..20); # G. C. Greubel, Mar 16 2020

CoefficientList[Series[-LambertW[-x/(1+x)], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
a[n_]:=If[n==1,1,n*Sum[Times@@a/@Length/@stn,{stn,Select[sps[Range[n-1]],Length[#]>1&]}]];
Array[a,10] (* Gus Wiseman, Dec 31 2019 *)

{ for (n=0, 100, f=n!; a=sum(k=1, n, (-1)^(n - k)*f/k!*binomial(n - 1, k - 1)*k^(k - 1)); write("b060356.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 04 2009

my(x='x+O('x^20)); concat([0], Vec(serlaplace(-lambertw(-x/(1+x))))) \\ G. C. Greubel, Feb 19 2018

a(n) = Sum_{k=1..n} (-1)^(n-k)*n!/k!*binomial(n-1, k-1)*k^(k-1). a(n) = Sum_{k=0..n} Stirling1(n, k)*A058863(k). - Vladeta Jovovic, Sep 17 2003
a(n) ~ n^(n-1) * (1-exp(-1))^(n+1/2). - Vaclav Kotesovec, Nov 27 2012
a(n) = n * A108919(n). - Gus Wiseman, Dec 31 2019

A305276 Expansion of e.g.f. 1/(1 + LambertW(-x/(1 - x))).

Original entry on oeis.org

1, 1, 6, 57, 748, 12565, 257526, 6232765, 173980920, 5502613833, 194477548330, 7596028355641, 324920533473108, 15106155118606045, 758463525318426942, 40901033617318501845, 2357682497456804486896, 144670077586483815863569, 9414952083720893890165842, 647715776085173413399687633

Offset: 0

Ilya Gutkovskiy, Aug 18 2018

nonn

Lah transform of A000312.

Robert Israel, Table of n, a(n) for n = 0..369
N. J. A. Sloane, Transforms

Cf. A000312, A052871, A060356, A305304.

S:= series(1/(1+LambertW(-x/(1-x))),x,51):
seq(coeff(S,x,j)*j!,j=0..50); # Robert Israel, Aug 19 2018

nmax = 19; CoefficientList[Series[1/(1 + LambertW[-x/(1 - x)]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[Binomial[n - 1, k - 1] k^k n!/k!, {k, n}], {n, 19}]]

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

a(n) ~ n^n * (1 + exp(1))^(n - 1/2) / exp(n - 1/2). - Vaclav Kotesovec, Aug 18 2018

A331726 E.g.f.: -LambertW(-x/(1 - x)) / (1 - x).

Original entry on oeis.org

0, 1, 6, 45, 448, 5825, 95796, 1926043, 45944256, 1269187137, 39840825700, 1400286658331, 54462564354672, 2321934762267601, 107664031299459012, 5393893268767761675, 290341440380472614656, 16710435419661861992705, 1024009456958258244673860

Offset: 0

Ilya Gutkovskiy, Jan 25 2020

nonn

Cf. A000169, A052871, A245496, A277505, A331727.

nmax = 18; CoefficientList[Series[-LambertW[-x/(1 - x)]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k]^2 k! (n - k)^(n - k - 1), {k, 0, n - 1}], {n, 0, 18}]

seq(n)={Vec(serlaplace(-lambertw(-x/(1 - x) + O(x*x^n)) / (1 - x)), -(n+1))} \\ Andrew Howroyd, Jan 25 2020

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

a(n) ~ (1 + exp(-1))^(n + 3/2) * n^(n-1). - Vaclav Kotesovec, Jan 26 2020

A376100 Expansion of e.g.f. -LambertW(-x / (1 - 2*x)).

Original entry on oeis.org

0, 1, 6, 57, 760, 13265, 289116, 7600873, 234730224, 8340307137, 335388171700, 15062758093361, 747393408423432, 40606032733746961, 2397539426985311532, 152864047998089113785, 10467226142002168282336, 766094017043351707135745

Offset: 0

Seiichi Manyama, Sep 10 2024

nonn

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

Cf. A052871, A376101.

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

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

E.g.f. A(x) satisfies A(x) = x * (2*A(x) + exp(A(x))).
E.g.f.: Series_Reversion( x / (2*x + exp(x)) ).
a(n) = n! * Sum_{k=1..n} 2^(n-k) * k^(k-1) * binomial(n-1,k-1)/k!.
a(n) ~ (1 + 2*exp(-1))^(n + 1/2) * n^(n-1). - Vaclav Kotesovec, Sep 10 2024

A376101 Expansion of e.g.f. -LambertW(-x / (1 - 3*x)).

Original entry on oeis.org

0, 1, 8, 99, 1684, 36865, 994986, 32106655, 1209994808, 52281293697, 2551380861070, 138903509144191, 8350198884092484, 549502839975044449, 39295464010757324930, 3034457861009541582015, 251666093876245502584816, 22310882229970705663827457

Offset: 0

Seiichi Manyama, Sep 10 2024

nonn

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

Cf. A052871, A376100.

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

a(n) = n!*sum(k=1, n, 3^(n-k)*k^(k-1)*binomial(n-1, k-1)/k!);

E.g.f. A(x) satisfies A(x) = x * (3*A(x) + exp(A(x))).
E.g.f.: Series_Reversion( x / (3*x + exp(x)) ).
a(n) = n! * Sum_{k=1..n} 3^(n-k) * k^(k-1) * binomial(n-1,k-1)/k!.
a(n) ~ (1 + 3*exp(-1))^(n + 1/2) * n^(n-1). - Vaclav Kotesovec, Sep 10 2024

A323772 Expansion of e.g.f. 1 - LambertW(-x/(1 - x))*(2 + LambertW(-x/(1 - x)))/2.

Original entry on oeis.org

1, 1, 3, 15, 112, 1165, 15966, 275149, 5743032, 141020793, 3984082570, 127298787121, 4538547029556, 178610366328277, 7690287949961358, 359592884584517445, 18146340023779538416, 982966789391874234865, 56889414275458791370770, 3503393307156206473624153, 228738978280736413137020460

Offset: 0

Ilya Gutkovskiy, Jan 27 2019

nonn

Cf. A000272, A052871, A305276.

seq(n!*coeff(series(1-LambertW(-x/(1-x))*(2+LambertW(-x/(1-x)))/2,x=0,21),x,n),n=0..20); # Paolo P. Lava, Jan 29 2019

nmax = 20; CoefficientList[Series[1 - LambertW[-x/(1 - x)] (2 + LambertW[-x/(1 - x)])/2, {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[Binomial[n - 1, k - 1] k^(k - 2) n!/k!, {k, n}], {n, 20}]]

a(n) = Sum_{k=0..n} binomial(n-1,k-1)*A000272(k)*n!/k!.

a(n) ~ (1 + exp(-1))^(n + 3/2) * n^(n-2). - Vaclav Kotesovec, Jan 27 2019

A376098 Expansion of e.g.f. -LambertW(-2*x / (1 - x))/2.

Original entry on oeis.org

0, 1, 6, 66, 1112, 25640, 753552, 26950000, 1136106624, 55167345792, 3032389917440, 186130732899584, 12619351769121792, 936591263680543744, 75527892444435486720, 6575887645386829301760, 614790327790529665138688, 61429094739085165675446272

Offset: 0

Seiichi Manyama, Sep 10 2024

nonn

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

Cf. A052871, A376099.

Cf. A352448, A376093.

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

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

E.g.f. A(x) satisfies A(x) = x * (A(x) + exp(2*A(x))).
E.g.f.: Series_Reversion( x / (x + exp(2*x)) ).
a(n) = n! * Sum_{k=1..n} (2*k)^(k-1) * binomial(n-1,k-1)/k!.
a(n) = n * A352448(n-1).
a(n) ~ (2 + exp(-1))^(n + 1/2) * n^(n-1) / 2^(3/2). - Vaclav Kotesovec, Sep 10 2024

A376099 Expansion of e.g.f. -LambertW(-3*x / (1 - x))/3.

Original entry on oeis.org

0, 1, 8, 123, 2940, 96465, 4035438, 205395687, 12320780328, 851216818977, 66565617543450, 5812559883272439, 560602050420898764, 59186681025383491281, 6789351417468526481526, 840843424588323640992615, 111820607202879512913388752, 15892724010727366554445999425

Offset: 0

Seiichi Manyama, Sep 10 2024

nonn

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

Cf. A052871, A376098.

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

a(n) = n!*sum(k=1, n, (3*k)^(k-1)*binomial(n-1, k-1)/k!);

E.g.f. A(x) satisfies A(x) = x * (A(x) + exp(3*A(x))).
E.g.f.: Series_Reversion( x / (x + exp(3*x)) ).
a(n) = n! * Sum_{k=1..n} (3*k)^(k-1) * binomial(n-1,k-1)/k!.
a(n) ~ (3 + exp(-1))^(n + 1/2) * n^(n-1) / 3^(3/2). - Vaclav Kotesovec, Sep 10 2024

A305304 Expansion of e.g.f. 1/(1 + LambertW(-x/(1 + x))).

Original entry on oeis.org

1, 1, 2, 9, 52, 405, 3786, 42301, 542984, 7924041, 129110230, 2327399481, 45940938924, 986045445853, 22856850513602, 569163515043285, 15150885843083536, 429364157810169105, 12905794670246364078, 410108007771441394129, 13736898888997174964660, 483740530150449507164901, 17866185834825657429606682

Offset: 0

Ilya Gutkovskiy, Aug 18 2018

nonn

Inverse Lah transform of A000312.

N. J. A. Sloane, Transforms

Cf. A000312, A052871, A060356, A305276.

a:=series(1/(1+LambertW(-x/(1+x))),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019

nmax = 22; CoefficientList[Series[1/(1 + LambertW[-x/(1 + x)]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] k^k n!/k!, {k, n}], {n, 22}]]

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

a(n) ~ n^n * (exp(1) - 1)^(n - 1/2) / exp(n - 1/2). - Vaclav Kotesovec, Aug 18 2018

A376327 Expansion of e.g.f. -LambertW(-x/(1-x^2)).

Original entry on oeis.org

0, 1, 2, 15, 112, 1285, 17616, 299299, 5946368, 136497897, 3544641280, 102858065431, 3297199331328, 115730076038317, 4414151526557696, 181797547951527915, 8040649885153755136, 380100842138029431121, 19125314442962053300224, 1020539634854353310016415, 57563650890815727219507200

Offset: 0

Vaclav Kotesovec, Sep 20 2024

nonn

Cf. A052871, A376106.

nmax=25; CoefficientList[Series[-LambertW[-x/(1-x^2)], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) ~ (1 + 4*exp(-2))^(1/4) * 2^n * n^(n-1) / (exp(n) * (sqrt(4 + exp(2)) - exp(1))^n).

cp's OEIS Frontend

A060356 Expansion of e.g.f.: -LambertW(-x/(1+x)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

PARI

Formula

A305276 Expansion of e.g.f. 1/(1 + LambertW(-x/(1 - x))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A331726 E.g.f.: -LambertW(-x/(1 - x)) / (1 - x).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A376100 Expansion of e.g.f. -LambertW(-x / (1 - 2*x)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A376101 Expansion of e.g.f. -LambertW(-x / (1 - 3*x)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A323772 Expansion of e.g.f. 1 - LambertW(-x/(1 - x))*(2 + LambertW(-x/(1 - x)))/2.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

A376098 Expansion of e.g.f. -LambertW(-2*x / (1 - x))/2.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A376099 Expansion of e.g.f. -LambertW(-3*x / (1 - x))/3.

Views

Author

Keywords

Links