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

%I A053492 #85 Jul 30 2025 11:24:55
%S A053492 1,2,15,184,3155,69516,1871583,59542064,2185497819,90909876100,
%T A053492 4226300379983,217152013181544,12219893000227107,747440554689309404,
%U A053492 49374719534173925055,3503183373320829575008,265693897270211120103563,21451116469521758657525748
%N A053492 REVEGF transform of [1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, ...].
%C A053492 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
%C A053492 a(n) is also the number of Schroeder trees on n vertices. - _Brad R. Jones_, May 09 2014
%C A053492 Number of pointed trees on pointed sets k[1...k...n] for any point k. - _Gus Wiseman_, Sep 27 2015
%H A053492 G. C. Greubel, <a href="/A053492/b053492.txt">Table of n, a(n) for n = 1..355</a>
%H A053492 W. Y. Chen, <a href="https://doi.org/10.1073/pnas.87.24.9635">A general bijective algorithm for trees</a>, PNAS December 1, 1990 vol. 87 no. 24 9635-9639.
%H A053492 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=854">Encyclopedia of Combinatorial Structures 854</a>.
%H A053492 Elena L. Wang and Guoce Xin, <a href="https://arxiv.org/abs/2507.15654">On Ward Numbers and Increasing Schröder Trees</a>, arXiv:2507.15654 [math.CO], 2025. See p. 12.
%F A053492 E.g.f. is the compositional inverse of 2*x - x*exp(x). - _Brian Drake_, Apr 25 2006
%F A053492 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
%F A053492 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
%F A053492 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
%F A053492 O.g.f.: x*Sum_{n>=0}  1/(2 - n*x)^(n+1). - _Paul D. Hanna_, Oct 27 2014
%F A053492 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
%e A053492 E.g.f.: A(x) = x + 2*x^2/2! + 15*x^3/3! + 184*x^4/4! + 3155*x^5/5! + ...
%e A053492 Related expansions from _Paul D. Hanna_, Jul 07 2012: (Start)
%e A053492 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! + ...
%e A053492 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)
%e A053492 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].
%p A053492 A:= series(RootOf(exp(_Z)*_Z+x-2*_Z), x, 30): A053492:= n-> n! * coeff(A, x, n); # _Brian Drake_, Apr 25 2006
%t A053492 Rest[CoefficientList[InverseSeries[Series[2*x-x*E^x, {x, 0, 20}], x],x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Oct 27 2014 *)
%o A053492 (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 */
%o A053492 (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 */
%o A053492 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A053492 {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
%o A053492 for(n=1, 25, print1(a(n), ", "))
%o A053492 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A053492 {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
%o A053492 for(n=1, 25, print1(a(n), ", "))
%o A053492 (PARI) \p100 \\ set precision
%o A053492 {A=Vec(sum(n=0, 400, 1./(2 - n*x +O(x^25))^(n+1)) )}
%o A053492 for(n=1, #A, print1(round(A[n]), ", ")) \\ _Paul D. Hanna_, Oct 27 2014
%Y A053492 Cf. A000311, A029768, A052894, A262673.
%K A053492 nonn
%O A053492 1,2
%A A053492 _N. J. A. Sloane_, Jan 15 2000
%E A053492 Signs removed by _Michael Somos_, based on _Brian Drake_'s remark, Jun 06 2012