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A053492 REVEGF transform of [1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, ...]. 29
1, 2, 15, 184, 3155, 69516, 1871583, 59542064, 2185497819, 90909876100, 4226300379983, 217152013181544, 12219893000227107, 747440554689309404, 49374719534173925055, 3503183373320829575008, 265693897270211120103563, 21451116469521758657525748 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

This sequence was incorrectly labeled. The REVEGF transform of [1, -1, -1, -1, -1, -1, ...], the sequence whose exponential generating function is the compositional inverse of 2*x - x*exp(x), is this sequence with all positive signs. - Brian Drake, Jun 16 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

LINKS

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

FORMULA

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

EXAMPLE

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].

MAPLE

A:= series(RootOf(exp(_Z)*_Z+x-2*_Z), x, 30): A053492:= n-> n! * coeff(A, x, n); # Brian Drake, Apr 25 2006

MATHEMATICA

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

PROG

(Maxima) 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

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

(PARI) {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), ", "))

(PARI) {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), ", "))

(PARI) \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

CROSSREFS

Cf. A000311, A029768, A052894, A262673.

Sequence in context: A196792 A210655 A052857 * A319834 A268420 A208402

Adjacent sequences:  A053489 A053490 A053491 * A053493 A053494 A053495

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jan 15 2000

EXTENSIONS

Signs removed by Michael Somos, Jun 06 2012

STATUS

approved

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Last modified September 30 13:46 EDT 2020. Contains 337439 sequences. (Running on oeis4.)