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%I M0796 #113 Apr 06 2022 08:17:34
%S 0,1,1,1,2,3,6,12,25,52,113,247,548,1226,2770,6299,14426,33209,76851,
%T 178618,416848,976296,2294224,5407384,12780394,30283120,71924647,
%U 171196956,408310668,975662480,2335443077,5599508648,13446130438,32334837886,77863375126,187737500013,453203435319,1095295264857,2649957419351
%N Number of rooted identity trees with n nodes (rooted trees whose automorphism group is the identity group).
%C The nodes are unlabeled.
%C There is a natural correspondence between rooted identity trees and finitary sets (sets whose transitive closure is finite); each node represents a set, with the children of that node representing the members of that set. When the set corresponding to an identity tree is written out using braces, there is one set of braces for each node of the tree; thus a(n) is also the number of sets that can be made using n pairs of braces. - _Franklin T. Adams-Watters_, Oct 25 2011.
%C Shifts left under WEIGH transform. - _Franklin T. Adams-Watters_, Jan 17 2007
%C Is this the sequence mentioned in the middle of page 355 of Motzkin (1948)? - _N. J. A. Sloane_, Jul 04 2015. Answer from _David Broadhurst_, Apr 06 2022: The answer is No. Motzkin was considering a sequence asymptotic to Catalan(n)/(4*n), namely A006082, which begins 1, 1, 1, 2, 3, 6, 12, 27, ... but he miscalculated and got 1, 1, 1, 2, 3, 6, 12, 25, ... instead! - _N. J. A. Sloane_, Apr 06 2022
%D F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 330.
%D S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 301 and 562.
%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 64, Eq. (3.3.15); p. 80, Problem 3.10.
%D D. E. Knuth, Fundamental Algorithms, 3rd Ed., 1997, pp. 386-388.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A004111/b004111.txt">Table of n, a(n) for n = 0..2500</a> (first 201 terms from T. D. Noe)
%H Joerg Arndt, <a href="/A004111/a004111.txt">all identity trees for n=1..11</a>
%H P. J. Cameron, <a href="http://dx.doi.org/10.1016/0012-365X(89)90081-2">Some sequences of integers</a>, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
%H A. Genitrini, <a href="http://arxiv.org/abs/1605.00837">Full asymptotic expansion for Polya structures</a>, arXiv:1605.00837 [math.CO], May 03 2016, p. 8.
%H Bernhard Gittenberger, Emma Yu Jin, Michael Wallner, <a href="https://arxiv.org/abs/1707.02144">On the shape of random Pólya structures</a>, arXiv|1707.02144 [math.CO], 2017-2018; Discrete Math., 341 (2018), 896-911.
%H Frank Harary and Geert Prins, <a href="http://dx.doi.org/10.1007/BF02559543">The number of homeomorphically irreducible trees and other species</a>, Acta Math., 101 (1959), 141-162.
%H F. Harary, R. W. Robinson and A. J. Schwenk, <a href="http://dx.doi.org/10.1017/S1446788700016190">Twenty-step algorithm for determining the asymptotic number of trees of various species</a>, J. Austral. Math. Soc., Series A, 20 (1975), 483-503.
%H F. Harary, R. W. Robinson and A. J. Schwenk, <a href="http://dx.doi.org/10.1017/S1446788700033760">Corrigenda: Twenty-step algorithm for determining the asymptotic number of trees of various species</a>, J. Austral. Math. Soc., Series A 41 (1986), p. 325.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=56">Encyclopedia of Combinatorial Structures 56</a>
%H P. Leroux and B. Miloudi, <a href="/A000081/a000081_2.pdf">Généralisations de la formule d'Otter</a>, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
%H T. Motzkin, <a href="http://dx.doi.org/10.1090/S0002-9904-1945-08486-9">The hypersurface cross ratio</a>, Bull. Amer. Math. Soc., 51 (1945), 976-984.
%H T. S. Motzkin, <a href="http://dx.doi.org/10.1090/S0002-9904-1948-09002-4 ">Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products</a>, Bull. Amer. Math. Soc., 54 (1948), 352-360.
%H N. J. A. Sloane, <a href="/A004111/a004111_1.pdf">Sketch showing trees with 2 through 6 nodes</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F Recurrence: a(n+1) = (1/n) * sum_{k=1..n} ( sum_{d|k} (-1)^(k/d+1) d*a(d) ) * a(n-k+1). - Mitchell Harris, Dec 02 2004
%F G.f. satisfies A(x) = x exp(A(x)-A(x^2)/2+A(x^3)/3-A(x^4)/4+...) [Harary and Prins]
%F Also A(x) = Sum_{n >= 1} a(n)*x^n = x * Product_{n >= 1} (1+x^n)^a(n).
%F a(n) ~ c * d^n / n^(3/2), where d = A246169 = 2.51754035263200389079535..., c = 0.3625364233974198712298411097408713812865256408189512533230825639621448038... . - _Vaclav Kotesovec_, Aug 22 2014, updated Dec 26 2020
%e The 2 identity trees with 4 nodes are:
%e O O
%e / \ |
%e O O O
%e | |
%e O O
%e |
%e O
%e These correspond to the sets {{},{{}}} and {{{{}}}}.
%e G.f.: x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 12*x^7 + 25*x^8 + 52*x^9 + ...
%p A004111 := proc(n)
%p spec := [ A, {A=Prod(Z,PowerSet(A))} ]:
%p combstruct[count](spec, size=n) ;
%p end proc:
%p # second Maple program:
%p with(numtheory):
%p a:= proc(n) a(n):= `if`(n<2, n, add(a(n-k)*add(a(d)*d*
%p (-1)^(k/d+1), d=divisors(k)), k=1..n-1)/(n-1))
%p end:
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Jul 15 2014
%t s[ n_, k_ ] := s[ n, k ]=a[ n+1-k ]+If[ n<2k, 0, -s[ n-k, k ] ]; a[ 1 ]=1; a[ n_ ] := a[ n ]=Sum[ a[ i ]s[ n-1, i ]i, {i, 1, n-1} ]/(n-1); Table[ a[ i ], {i, 1, 30} ] (* _Robert A. Russell_ *)
%t a[ n_] := If[ n < 2, Boole[n == 1], Nest[ CoefficientList[ Normal[ Times @@ (Table[1 + x^k, {k, Length@#}]^#) + x O[x]^Length@#], x] &, {}, n - 1][[n]]]; (* _Michael Somos_, Jul 10 2014 *)
%t a[n_] := a[n] = Sum[a[n-k]*Sum[a[d]*d*(-1)^(k/d+1),{d, Divisors[k]}], {k, 1, n-1}]/(n-1); a[0]=0; a[1]=1; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Feb 02 2015 *)
%o (Haskell)
%o import Data.List (genericIndex)
%o a004111 = genericIndex a004111_list
%o a004111_list = 0 : 1 : f 1 [1] where
%o f x zs = y : f (x + 1) (y : zs) where
%o y = (sum $ zipWith (*) zs $ map g [1..]) `div` x
%o g k = sum $ zipWith (*) (map (((-1) ^) . (+ 1)) $ reverse divs)
%o (zipWith (*) divs $ map a004111 divs)
%o where divs = a027750_row k
%o -- _Reinhard Zumkeller_, Apr 29 2014
%o (PARI)
%o N=66; A=vector(N+1, j, 1);
%o for (n=1, N, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1) * d * A[d]) * A[n-k+1] ) );
%o concat([0], A)
%o \\ _Joerg Arndt_, Jul 10 2014
%Y Cf. A000009, A000081, A000220, A196118, A196154, A196161, A227819.
%Y Cf. A027750, A035056, A246169.
%Y Column k=1 of A255517, A316074, A316101, A318757.
%K nonn,easy,nice,eigen
%O 0,5
%A _N. J. A. Sloane_
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