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A053492
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REVEGF transform of [1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, ...].
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31
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1, 2, 15, 184, 3155, 69516, 1871583, 59542064, 2185497819, 90909876100, 4226300379983, 217152013181544, 12219893000227107, 747440554689309404, 49374719534173925055, 3503183373320829575008, 265693897270211120103563, 21451116469521758657525748
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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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
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EXAMPLE
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E.g.f.: A(x) = x + 2*x^2/2! + 15*x^3/3! + 184*x^4/4! + 3155*x^5/5! + ...
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].
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MAPLE
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A:= series(RootOf(exp(_Z)*_Z+x-2*_Z), x, 30): A053492:= n-> n! * coeff(A, x, n); # Brian Drake, Apr 25 2006
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MATHEMATICA
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Rest[CoefficientList[InverseSeries[Series[2*x-x*E^x, {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Oct 27 2014 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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