A052894 -id:A052894 - cp's OEIS frontend

A053492 REVEGF transform of [1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, ...].

1, 2, 15, 184, 3155, 69516, 1871583, 59542064, 2185497819, 90909876100, 4226300379983, 217152013181544, 12219893000227107, 747440554689309404, 49374719534173925055, 3503183373320829575008, 265693897270211120103563, 21451116469521758657525748

Offset: 1

N. J. A. Sloane, Jan 15 2000

nonn

Sequence gives the number of total circled partitions of n. This is the number of ways to partition n into at least two blocks, circle one block, then successively partition each non-singleton block into at least two blocks and circle one of the blocks. Stop when only singleton blocks remain. - Brian Drake, Apr 25 2006
a(n) is also the number of Schroeder trees on n vertices. - Brad R. Jones, May 09 2014
Number of pointed trees on pointed sets k[1...k...n] for any point k. - Gus Wiseman, Sep 27 2015

			E.g.f.: A(x) = x + 2*x^2/2! + 15*x^3/3! + 184*x^4/4! + 3155*x^5/5! + ...
Related expansions from _Paul D. Hanna_, Jul 07 2012: (Start)
A(x) = x + (exp(x)-1)*x + d/dx (exp(x)-1)^2*x^2/2! + d^2/dx^2 (exp(x)-1)^3*x^3/3! + d^3/dx^3 (exp(x)-1)^4*x^4/4! + ...
log(A(x)/x) = (exp(x)-1) + d/dx (exp(x)-1)^2*x/2! + d^2/dx^2 (exp(x)-1)^3*x^2/3! + d^3/dx^3 (exp(x)-1)^4*x^3/4! + ... (End)
The a(3) = 15 pointed trees are 1[1 2[2 3]], 1[1 3[2 3]], 1[1[1 3] 2], 1[1[1 2] 3], 1[1 2 3], 2[1 2[2 3]], 2[1[1 3] 2], 2[2 3[1 3]], 2[2[1 2] 3], 2[1 2 3], 3[1 3[2 3]], 3[2 3[1 3]], 3[1[1 2] 3], 3[2[1 2] 3], 3[1 2 3].

G. C. Greubel, Table of n, a(n) for n = 1..355
W. Y. Chen, A general bijective algorithm for trees, PNAS December 1, 1990 vol. 87 no. 24 9635-9639.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 854.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 12.

Cf. A000311, A029768, A052894, A262673.

A:= series(RootOf(exp(A053492:=%20n-%3E%20n!%20*%20coeff(A,%20x,%20n);%20%23%20_Brian%20Drake">Z)*_Z+x-2*_Z), x, 30): A053492:= n-> n! * coeff(A, x, n); # _Brian Drake, Apr 25 2006

Rest[CoefficientList[InverseSeries[Series[2*x-x*E^x, {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Oct 27 2014 *)

a(n):= if n=1 then 1 else sum(k!*stirling2(n-1,k)*binomial(n+k-1,n-1),k,1,n-1); /* Vladimir Kruchinin, May 10 2011 */

{a(n) = if( n<1, 0, n! * polcoeff( serreverse( 2*x - x * exp(x + x * O(x^n))), n))}; /* Michael Somos, Jun 06 2012 */

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, (exp(x+x*O(x^n))-1)^m*x^m/m!)); n!*polcoeff(A, n)} \\ Paul D. Hanna, Jul 07 2012
for(n=1, 25, print1(a(n), ", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, (exp(x+x*O(x^n))-1)^m*x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)} \\ Paul D. Hanna, Jul 07 2012
for(n=1, 25, print1(a(n), ", "))

\p100 \\ set precision
{A=Vec(sum(n=0, 400, 1./(2 - n*x +O(x^25))^(n+1)) )}
for(n=1, #A, print1(round(A[n]), ", ")) \\ Paul D. Hanna, Oct 27 2014

E.g.f. is the compositional inverse of 2*x - x*exp(x). - Brian Drake, Apr 25 2006
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) (exp(x)-1)^n*x^n / n!. - Paul D. Hanna, Jul 07 2012
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (exp(x)-1)^n*x^(n-1) / n! ). - Paul D. Hanna, Jul 07 2012
a(n) = Sum_{k=1..n-1} k!*Stirling2(n-1,k)*C(n+k-1,n-1), n > 1, a(1)=1. - Vladimir Kruchinin, May 10 2011
O.g.f.: x*Sum_{n>=0}  1/(2 - n*x)^(n+1). - Paul D. Hanna, Oct 27 2014
a(n) ~ n^(n-1) * (LambertW(2*exp(1)))^n / (sqrt(1+LambertW(2*exp(1))) * 2^n * exp(n) * (LambertW(2*exp(1))-1)^(2*n-1)). - Vaclav Kotesovec, Oct 27 2014

Signs removed by Michael Somos, based on Brian Drake's remark, Jun 06 2012

A052802 E.g.f. satisfies A(x) = 1/(1 + log(1 - x*A(x))).

Original entry on oeis.org

1, 1, 5, 47, 660, 12414, 293552, 8374806, 280064600, 10747277832, 465597887592, 22479948822792, 1197060450322800, 69699159437088960, 4405397142701855232, 300408348609092268144, 21983809533066553697280

Offset: 0

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

Previous name was: A simple grammar.

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 47*x^3/3! + 660*x^4/4! +... [_Paul D. Hanna_, Aug 28 2008]

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 760

Cf. A052819. [From Paul D. Hanna, Aug 28 2008]

Cf. A052894, A198860.

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

CoefficientList[1/x*InverseSeries[Series[x + x*Log[1-x], {x, 0, 20}], x],x] * Range[0, 19]! (* Vaclav Kotesovec, Jan 08 2014 *)

a(n):=sum(binomial(n+k,n)*k!*sum((-1)^(n+j)/(k-j)!*sum(stirling1(n,j-i)/i!,i,0,j),j,0,k),k,0,n)/(n+1); /* Vladimir Kruchinin, May 09 2013 */

a(n)=n!*polcoeff((1/x)*serreverse(x+x*log(1-x +x*O(x^n))),n) \\ Paul D. Hanna, Aug 28 2008

From Paul D. Hanna, Aug 28 2008: (Start)
E.g.f. satisfies: A(x*(1 + log(1-x))) = 1/(1 + log(1-x)).
E.g.f. satisfies: A(x) = 1/(1 + log(1 - x*A(x))).
E.g.f.: A(x) = (1/x)*Series_Reversion[x + x*log(1-x)]. (End)
a(n)=sum(k=0..n, binomial(n+k,n)*k!*sum(j=0..k, (-1)^(n+j)/(k-j)!*sum(i=0..j, stirling1(n,j-i)/i!)))/(n+1); [Vladimir Kruchinin, May 09 2013]
a(n) ~ n^(n-1) * c^n / (sqrt(1+c) * exp(n) * (c-1)^(2*n+1)), where c = LambertW(exp(2)) = 1.5571455989976114... - Vaclav Kotesovec, Jan 08 2014
a(n) = (1/(n+1)!) * Sum_{k=0..n} (n+k)! * |Stirling1(n,k)|. - Seiichi Manyama, Nov 06 2023

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

A370988 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 - x(exp(x) - 1)) ).

Original entry on oeis.org

1, 0, 2, 3, 76, 425, 10326, 119077, 3158968, 57929265, 1740086290, 44066266541, 1512768107940, 48660920528233, 1905202422005806, 73878129769929045, 3275941116578461936, 147981592692778718561, 7366814796135956094378, 378666415166758834858237

Offset: 0

Seiichi Manyama, Mar 06 2024

nonn

Index entries for reversions of series

Cf. A052894, A370989, A370990.
Cf. A370991, A370992.
Cf. A052848.

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

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

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (n+k)! * Stirling2(n-k,k)/(n-k)!.

A367134 E.g.f. satisfies A(x) = 1/(2 - exp(x*A(x)^2)).

Original entry on oeis.org

1, 1, 7, 97, 2051, 58681, 2122695, 92960001, 4782826459, 282821367001, 18901822316543, 1409070858589153, 115925274671836371, 10433564954705754681, 1019782291631652745591, 107570331041074850633473, 12180277895590328004331019, 1473587743517654702900335705

Offset: 0

Seiichi Manyama, Nov 06 2023

nonn

Cf. A000670, A052894, A367135.

Cf. A367136, A367138.

Table[1/(2*n+1)! * Sum[(2*n+k)! * StirlingS2[n,k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 07 2023 *)

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

a(n) = (1/(2*n+1)!) * Sum_{k=0..n} (2*n+k)! * Stirling2(n,k).

a(n) ~ 2^(n-1) * LambertW(exp(1/2))^(2*n + 1) * n^(n-1) / (sqrt(1 + LambertW(exp(1/2))) * exp(n) * (2*LambertW(exp(1/2)) - 1)^(3*n + 1)). - Vaclav Kotesovec, Nov 07 2023

A367135 E.g.f. satisfies A(x) = 1/(2 - exp(x*A(x)^3)).

Original entry on oeis.org

1, 1, 9, 166, 4719, 182326, 8927301, 529922002, 36988772211, 2969132797966, 269488306924833, 27291375956851546, 3050923148547318039, 373187615576953777510, 49580088565083198922845, 7109665420655116517351458, 1094492388113416460752513851

Offset: 0

Seiichi Manyama, Nov 06 2023

nonn

Cf. A000670, A052894, A367134.

Cf. A367137, A367139.

Table[1/(3*n+1)! * Sum[(3*n+k)! * StirlingS2[n,k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 07 2023 *)

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

a(n) = (1/(3*n+1)!) * Sum_{k=0..n} (3*n+k)! * Stirling2(n,k).

a(n) ~ 3^(4*n) * LambertW(2*exp(1/3)/3)^(3*n + 1) * n^(n-1) / (sqrt(1 + LambertW(2*exp(1/3)/3)) * 2^(3*n + 1) * exp(n) * (3*LambertW(2*exp(1/3)/3) - 1)^(4*n + 1)). - Vaclav Kotesovec, Nov 07 2023

A258922 E.g.f. satisfies A(x) = 1/(3 - 2exp(xA(x))).

Original entry on oeis.org

1, 2, 18, 302, 7562, 253542, 10685794, 543309230, 32378850042, 2214215333750, 170939286647570, 14707184259036414, 1395561779648175274, 144795755972202587462, 16308198003201872476866, 1981633767850818093910094, 258406311809937562215099482, 35994776359231593721760238102

Offset: 0

Paul D. Hanna, Jun 18 2015

nonn

			E.g.f.: A(x) = 1 + 2*x + 18*x^2/2! + 302*x^3/3! + 7562*x^4/4! + 253542*x^5/5! +...
where A(3*x - 2*x*exp(x)) = 1/(3 - 2*exp(x)).

Cf. A259063, A052894, A258923.

CoefficientList[1/x*InverseSeries[Series[3*x - 2*x*E^x, {x, 0, 21}], x],x] * Range[0, 20]! (* Vaclav Kotesovec, Jun 19 2015 *)

{a(n) = local(A=1); A = (1/x)*serreverse(3*x - 2*x*exp(x +x^2*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=1); A = 1 + (1/x)*sum(m=1, n+1, Dx(m-1, 2^m*(exp(x+x*O(x^n))-1)^m * x^m/m!)); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=1+x+x*O(x^n)); A = exp(sum(m=1, n+1, Dx(m-1, 2^m*(exp(x+x*O(x^n))-1)^m * x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

E.g.f. A(x) satisfies:
(1) A(x) = (1/x) * Series_Reversion( 3*x - 2*x*exp(x) ).
(2) A(x) = 1 + (1/x) * Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (exp(x)-1)^n * x^n / n!.
(3) A(x) = exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (exp(x)-1)^n * x^(n-1) / n! ).
a(n) = A259063(n+1) / (n+1). - Vaclav Kotesovec, Jun 19 2015
a(n) ~ (c/3)^(n+1) * n^(n-1) / (sqrt(c+1) * exp(n) * (c-1)^(2*n+1)), where c = LambertW(3*exp(1)/2). - Vaclav Kotesovec, Jun 19 2015
a(n) = (1/(n+1)!) * Sum_{k=0..n} 2^k * (n+k)! * Stirling2(n,k). - Seiichi Manyama, Mar 06 2024

A262673 Number of pointed trees on normal pointed multisets of weight n.

Original entry on oeis.org

1, 3, 34, 615, 15410, 494018

Offset: 1

Gus Wiseman, Sep 26 2015

A pointed multiset is normal if its entries span an initial interval of positive integers.

			The a(2) = 3 pointed trees are 1[1 1], 1[1 2], 2[1 2].
The a(3) = 34 pointed trees of the form rootpoint[pointedbranch ... pointedbranch] are:
1[1 1[1 1]], 1[1[1 1] 1], 1[1 1 1],
1[1 2[2 2]], 1[1[1 2] 2], 1[1 2 2],
2[1 2[2 2]], 2[1[1 2] 2], 2[2 2[1 2]], 2[2[1 2] 2], 2[1 2 2],
1[1 1[1 2]], 1[1 2[1 2]], 1[1[1 1] 2], 1[1[1 2] 1], 1[1 1 2],
2[1 2[1 2]], 2[1[1 1] 2], 2[1 1 2],
1[1 2[2 3]], 1[1 3[2 3]], 1[1[1 2] 3], 1[1[1 3] 2], 1[1 2 3],
2[1 2[2 3]], 2[1[1 3] 2], 2[2 3[1 3]], 2[2[1 2] 3], 2[1 2 3],
3[1 3[2 3]], 3[1[1 2] 3], 3[2 3[1 3]], 3[2[1 2] 3], 3[1 2 3].

Cf. A053492, A052894, A262671.

A367162 E.g.f. satisfies A(x) = 1 + A(x)^2 * (exp(x*A(x)) - 1).

Original entry on oeis.org

1, 1, 7, 94, 1917, 52806, 1837511, 77372590, 3826454617, 217450806550, 13964683107195, 1000222945246878, 79058281093939109, 6835704081028512886, 641830800234353986639, 65035909964873069979598, 7073810997780630959477937, 822049309574436641341233366

Offset: 0

Seiichi Manyama, Nov 07 2023

nonn

Cf. A000272, A052894, A367163.

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

a(n) = Sum_{k=0..n} (n+2*k)!/(n+k+1)! * Stirling2(n,k).

A370989 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 - x^2(exp(x) - 1)) ).

Original entry on oeis.org

1, 0, 0, 6, 12, 20, 2910, 22722, 117656, 8482392, 143398170, 1519998590, 79655138772, 2206506673956, 39101112995126, 1798446230741370, 68667380639283120, 1795441154500375472, 81344029377887798706, 3830461514154681289974, 135388937631209203030700

Offset: 0

Seiichi Manyama, Mar 06 2024

nonn

Index entries for reversions of series

Cf. A052894, A370988, A370990.

Cf. A358013.

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

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

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} (n+k)! * Stirling2(n-2*k,k)/(n-2*k)!.

A377326 E.g.f. satisfies A(x) = 1 + (exp(x*A(x)) - 1)/A(x).

Original entry on oeis.org

1, 1, 1, 4, 15, 96, 665, 6028, 60907, 725560, 9591549, 142574004, 2323440119, 41519079616, 803667844993, 16797423268252, 376458083887875, 9014414549836296, 229564623594841637, 6197477089425914692, 176767174407208663759, 5312208220728020517136, 167760328500471584529321

Offset: 0

Seiichi Manyama, Oct 24 2024

Cf. A052894, A367162, A367163.

Cf. A367180, A377324.

terms=23; A[]=1; Do[A[x] = 1 + (Exp[x*A[x]] - 1)/A[x]+ O[x]^terms // Normal, terms]; CoefficientList[Series[A[x],{x,0,terms}],x]Range[0,terms-1]! (* Stefano Spezia, Aug 28 2025 *)

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

a(n) = Sum_{k=0..floor((n+1)/2)} (n-k)!/(n-2*k+1)! * Stirling2(n,k).

cp's OEIS Frontend

A053492 REVEGF transform of [1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, ...].

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A052802 E.g.f. satisfies A(x) = 1/(1 + log(1 - x*A(x))).

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A370988 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*(exp(x) - 1)) ).

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A367134 E.g.f. satisfies A(x) = 1/(2 - exp(x*A(x)^2)).

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A367135 E.g.f. satisfies A(x) = 1/(2 - exp(x*A(x)^3)).

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A258922 E.g.f. satisfies A(x) = 1/(3 - 2*exp(x*A(x))).

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A262673 Number of pointed trees on normal pointed multisets of weight n.

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A370988 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 - x(exp(x) - 1)) ).

A258922 E.g.f. satisfies A(x) = 1/(3 - 2exp(xA(x))).

A370989 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 - x^2(exp(x) - 1)) ).