A058866 -id:A058866 - cp's OEIS frontend

A005750 Number of planted matched trees with n nodes.

1, 1, 3, 10, 39, 160, 702, 3177, 14830, 70678, 342860, 1686486, 8393681, 42187148, 213828802, 1091711076, 5609297942, 28982708389, 150496728594, 784952565145, 4110491658233, 21602884608167, 113907912618599, 602414753753310, 3194684310627727, 16984594260224529

Offset: 1

N. J. A. Sloane

nonn

When convolved with itself gives A000151.
Number of rooted trees with n nodes and edges not attached to root are 2-colored or oriented.
Also number of 2-trees (with 2n+1 cells) rooted at a symmetric end-edge. - Vladeta Jovovic, Aug 22 2001

			A(x) = x + x^2 + 3*x^3 + 10*x^4 + 39*x^5 + 160*x^6 + 702*x^7 + ...

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6.5.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 75, Eq. (3.5.3).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 1..1325 (terms 1..500 from Alois P. Heinz)
Loïc Foissy, Algebraic structures on typed decorated rooted trees, arXiv:1811.07572 [math.RA], 2018.
T. Fowler, I. Gessel, G. Labelle, P. Leroux, The specification of 2-trees, Adv. Appl. Math. 28 (2) (2002) 145-168, Table 1.
Andrew Gainer-Dewar, Gamma-Species and the Enumeration of k-Trees, Electronic Journal of Combinatorics, Volume 19 (2012), #P45. See page 20, line -3. - From _N. J. A. Sloane_, Dec 15 2012
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 428
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)
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A000151, A058870, A058866, A054581, A245870.

A:= proc(n) option remember; if n=0 then 0 else unapply(convert(series(x*exp(add((A(n-1)(x^k))^2/(k*x^k), k=1..2*n)), x=0,2*n), polynom), x) fi end: a:= n-> coeff(series(A(n)(x), x=0, n+1), x,n): seq(a(n), n=1..23); # Alois P. Heinz, Aug 20 2008

max = 23; f[x_] := Sum[c[k]*x^k, {k, 0, max}]; c[0] = 0; c[1] = 1; coes = CoefficientList[ Series[ Log[f[x]/x] - Sum[f[x^k]^2/(k*x^k), {k, 1, max}], {x, 0, max}], x]; eqns = Rest[ Thread[coes == 0]]; s[2] = Solve[eqns[[1]], c[2]][[1]]; Do[eqns = Rest[eqns] /. s[k-1]; s[k] = Solve[ eqns[[1]], c[k]][[1]], {k, 3, max}]; Table[c[k], {k, 1, max}] /. Flatten[ Table[s[k], {k, 2, max}]] (* Jean-François Alcover, Oct 25 2011, after g.f. *)
terms = 26; (* B = g.f. of A000151 *) B[] = 0; Do[B[x] = x*Exp[2*Sum[ B[x^k]/k, {k, 1, terms}]] + O[x]^terms // Normal, terms];
A[x_] = Exp[Sum[B[x^k]/k, {k, 1, terms}]] + O[x]^terms;
CoefficientList[A[x], x] (* Jean-François Alcover, Jan 11 2018 *)

seq(N) = {my(A=vector(N, j, 1)); for(n=1, N-1, A[n+1] = 2/n * sum(i=1, n, sumdiv(i, d, d*A[d]) * A[n-i+1] ) ); Vec(sqrt(Ser(A)))} \\ Andrew Howroyd, May 13 2018

a(n+1) is Euler transform of A000151.
G.f.: A(x) = x*exp( A(x)^2/x + A(x^2)^2/(2x^2) + A(x^3)^2/(3x^3) + ... + A(x^n)^2/(n*x^n) + ...). [Harary & Palmer (3.5.8)] - Paul D. Hanna
G.f.: sqrt(B(x)/x) where B(x) is the g.f. of A000151. - Andrew Howroyd, May 13 2018
a(n) ~ c * d^n / n^(3/2), where d = A245870 = 5.646542616232..., c = 0.06185402386554883780092844840921448929211072031752507960399709674242810089... - Vaclav Kotesovec, Sep 12 2014, updated Dec 26 2020
a(n) = A063687(n)+2*A058870(n). [Harary & Palmer (3.5.3)] - R. J. Mathar, Jan 13 2025

More terms, formula and comment from Christian G. Bower, Dec 15 1999

A058870 Number of 2-trees rooted at an asymmetric edge.

Original entry on oeis.org

0, 1, 4, 18, 77, 346, 1578, 7396, 35297, 171352, 843067, 4196502, 21092793, 106912874, 545851964, 2804641873, 14491337393, 75248330560, 392476202012, 2055245665857, 10801441911431, 56953955507744, 301207374937558

Offset: 1

N. J. A. Sloane, Jan 06 2001

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 75, Eq. (3.5.10).

Cf. A058870, A058866, A054581.

Let A063687(n) be the number of 2-trees rooted at a symmetric edge (see H. and P. Eq. (3.5.9)). Then A058866(n) = A058870(n) + A063687(n).

H. and P. give g.f.

More terms from Vladeta Jovovic, Aug 22 2001

A063687 Number of 2-trees rooted at any symmetric edge.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 21, 38, 84, 156, 352, 677, 1562, 3054, 7148, 14196, 33603, 67474, 161121, 326519, 785305, 1603111, 3878194, 7965845, 19365811, 39983300, 97610247, 202424674, 495955345, 1032460679, 2537529717, 5300256499

Offset: 0

Vladeta Jovovic, Aug 22 2001

nonn

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 75, Eq. (3.5.9).

Cf. A058870, A058866, A054581.

A370771 Triangle read by rows: T(n,k) is the number of k-trees with n unlabeled nodes rooted at a front.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 1, 9, 6, 2, 1, 1, 1, 20, 21, 6, 2, 1, 1, 1, 48, 83, 25, 6, 2, 1, 1, 1, 115, 356, 126, 25, 6, 2, 1, 1, 1, 286, 1599, 745, 135, 25, 6, 2, 1, 1, 1, 719, 7434, 4784, 895, 135, 25, 6, 2, 1, 1, 1, 1842, 35381, 32372, 6846, 915, 135, 25, 6, 2, 1, 1

Offset: 0

Andrew Howroyd, Mar 01 2024

A front is a k-clique.

			Triangle begins:
  1;
  1,   1;
  1,   1,    1;
  1,   2,    1,    1;
  1,   4,    2,    1,   1;
  1,   9,    6,    2,   1,   1;
  1,  20,   21,    6,   2,   1,  1;
  1,  48,   83,   25,   6,   2,  1, 1;
  1, 115,  356,  126,  25,   6,  2, 1, 1;
  1, 286, 1599,  745, 135,  25,  6, 2, 1, 1;
  1, 719, 7434, 4784, 895, 135, 25, 6, 2, 1, 1;
  ...

Andrew Gainer-Dewar, Gamma-Species and the Enumeration of k-Trees, Electronic Journal of Combinatorics, Volume 19 (2012), #P45.

Columns k=0..2 are A000012, A000081, A058866.

Cf. A370770 (unrooted), A370772, A370773.

A063688 Number of 2-trees rooted at a triangle.

Original entry on oeis.org

1, 1, 3, 10, 39, 164, 746, 3474, 16658, 81166, 401169, 2004517, 10110757, 51402250, 263133142, 1355126922, 7016115632, 36498130908, 190673015083, 999932115039, 5262094054524, 27779114013628, 147072756065567, 780722981065006

Offset: 1

Vladeta Jovovic, Aug 22 2001

nonn

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 76, Eq. (3.5.13).

Cf. A005750, A058870, A058866, A054581.

A063682 Number of 2-trees rooted at an asymmetric end-edge.

Original entry on oeis.org

0, 1, 3, 13, 52, 229, 1024, 4749, 22454, 108299, 529896, 2625903, 13148268, 66428383, 338197609, 1733399552, 8936746984, 46315049443, 241146306914, 1260805087503, 6616786338256, 34843842405490, 184057255372576

Offset: 1

Vladeta Jovovic, Aug 22 2001

nonn

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 75, Eq. (3.5.4).

Cf. A058870, A058866, A054581.

A063689 Number of 2-trees rooted at a triangle with 3 similar edges.

Original entry on oeis.org

1, 1, 2, 6, 21, 83, 356, 1599, 7434, 35381, 171508, 843419, 4197179, 21094355, 106915928, 545859112, 2804656069, 14491370996, 75248398034, 392476363133, 2055245992376, 10801442696736, 56953957110855, 301207378815752, 1597342159296786, 8492297139795170

Offset: 1

Vladeta Jovovic, Aug 22 2001

nonn

			Sequence really begins 1, 0, 0, 1, 0, 0, 2, 0, 0, 6, 0, 0, 21, 0, 0, 83, 0, 0, 356, ... but only nonzero trisection is shown.

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 76, Eq. (3.5.17).

Cf. A005750, A058870, A058866, A054581.

a(n) = A058870(n) + A063687(n) with A058870(0)=0. - Sean A. Irvine, May 07 2023

More terms from Sean A. Irvine, May 07 2023

A063692 Number of 2-trees rooted at a triangle with two similar edges.

Original entry on oeis.org

1, 2, 2, 7, 11, 25, 43, 106, 180, 453, 797, 2023, 3632, 9328, 16960, 44036, 80989, 211815, 393098, 1034958, 1934813, 5121356, 9633260, 25615432, 48433926, 129289382, 245554773, 657691061, 1253974468, 3368475942, 6444250241

Offset: 2

Vladeta Jovovic, Aug 23 2001

nonn

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 76, Eq. (3.5.16).

Cf. A005750, A058870, A058866, A054581.

cp's OEIS Frontend

A005750 Number of planted matched trees with n nodes.

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A058870 Number of 2-trees rooted at an asymmetric edge.

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A063687 Number of 2-trees rooted at any symmetric edge.

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A370771 Triangle read by rows: T(n,k) is the number of k-trees with n unlabeled nodes rooted at a front.

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A063688 Number of 2-trees rooted at a triangle.

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A063682 Number of 2-trees rooted at an asymmetric end-edge.

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A063689 Number of 2-trees rooted at a triangle with 3 similar edges.

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A063692 Number of 2-trees rooted at a triangle with two similar edges.

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