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 A005753 Number of rooted identity matched trees with n nodes. (Formerly M1514) 7
 1, 2, 5, 18, 66, 266, 1111, 4792, 21124, 94888, 432415, 1994828, 9296712, 43706722, 207030398, 987130456, 4733961435, 22819241034, 110500644857, 537295738556, 2622248720234, 12840953621208, 63074566121245, 310693364823376, 1534374047239554, 7595642577152762 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Also number of rooted identity trees with n nodes and 2-colored non-root nodes. - Christian G. Bower, Apr 15 1998 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Alois P. Heinz, Table of n, a(n) for n = 1..1000 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 429 R. Simion, Trees with 1-factors and oriented trees, Discrete Math., 88 (1991), 93-104. R. Simion, Trees with 1-factors and oriented trees, Discrete Math., 88 (1981), 97. (Annotated scanned copy) FORMULA G.f.: x*Product_{n>=1} (1 + x^n)^(2*a(n)) = Sum_{n>=1} a(n)*x^n. - Paul D. Hanna, Dec 31 2011 a(n) ~ c * d^n / n^(3/2), where d = A246312 = 5.249032491228170579164952216..., c = 0.192066288645200371237879149260484794708740197522264442948290580404909605849... - Vaclav Kotesovec, Aug 25 2014, updated Dec 26 2020 G.f. A(x) satisfies: A(x) = x*exp(2*Sum_{k>=1} (-1)^(k+1)*A(x^k)/k). - Ilya Gutkovskiy, Apr 13 2019 EXAMPLE G.f.: A(x) = x + 2*x^2 + 5*x^3 + 18*x^4 + 66*x^5 + 266*x^6 + ... where A(x) = x*(1+x)^2*(1+x^2)^4*(1+x^3)^10*(1+x^4)^36*(1+x^5)^132*... (the exponents are A038077(n), n>=1). MAPLE b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,       add(binomial(2*a(i), j)*b(n-i*j, i-1), j=0..n/i)))     end: a:= n-> `if`(n=1, 1, b((n-1)\$2)): seq(a(n), n=1..40);  # Alois P. Heinz, Aug 01 2013 MATHEMATICA b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[2*a[i], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := If[n == 1, 1, b[n-1, n-1]]; Table[a[n] // FullSimplify, {n, 1, 30}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *) PROG (PARI) {a(n)=polcoeff(x*prod(k=1, n-1, (1+x^k+x*O(x^n))^(2*a(k))), n)} /* Paul D. Hanna */ CROSSREFS Cf. A038077, A246312. Column k=2 of A255517. Sequence in context: A150017 A150018 A150019 * A150020 A144721 A150021 Adjacent sequences:  A005750 A005751 A005752 * A005754 A005755 A005756 KEYWORD nonn,eigen AUTHOR STATUS approved

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Last modified May 14 07:13 EDT 2021. Contains 343879 sequences. (Running on oeis4.)