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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A000669 Number of series-reduced planted trees with n leaves. Also the number of essentially series series-parallel networks with n edges; also the number of essentially parallel series-parallel networks with n edges.

Original entry on oeis.org

1, 1, 2, 5, 12, 33, 90, 261, 766, 2312, 7068, 21965, 68954, 218751, 699534, 2253676, 7305788, 23816743, 78023602, 256738751, 848152864, 2811996972, 9353366564, 31204088381, 104384620070, 350064856815, 1176693361956, 3963752002320
Offset: 1

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Author

N. J. A. Sloane and John Riordan

Keywords

Comments

Also the number of unlabeled connected cographs on n nodes. - N. J. A. Sloane and Eric W. Weisstein, Oct 21 2003
A cograph is a simple graph which contains no path of length 3 as an induced subgraph. - Michael Somos, Apr 19 2014
Also called "hierarchies" by Genitrini (2016). - N. J. A. Sloane, Mar 24 2017

Examples

			G.f. = x + x^2 + 2*x^3 + 5*x^4 + 12*x^5 + 33*x^6 + 90*x^7 + 261*x^8 + ...
a(4)=5 with the following series-reduced planted trees: (oooo), (oo(oo)), (o(ooo)), (o(o(oo))), ((oo)(oo)). - _Michael Somos_, Jul 25 2003

References

  • N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, p. 43.
  • A. Brandstaedt, V. B. Le and J. P. Spinrad, Graph Classes: A Survey, SIAM Publications, 1999. (For definition of cograph)
  • A. Cayley, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 3, p. 246.
  • D. E. Knuth, The Art of Computer Programming, 3rd ed. 1997, Vol. 1, p. 589, Answers to Exercises Section 2.3.4.4 5.
  • L. F. Meyers, Corrections and additions to Tree Representations in Linguistics. Report 3, 1966, p. 138. Project on Linguistic Analysis, Ohio State University Research Foundation, Columbus, Ohio.
  • L. F. Meyers and W. S.-Y. Wang, Tree Representations in Linguistics. Report 3, 1963, pp. 107-108. Project on Linguistic Analysis, Ohio State University Research Foundation, Columbus, Ohio.
  • J. Riordan and C. E. Shannon, The number of two-terminal series-parallel networks, J. Math. Phys., 21 (1942), 83-93 (the numbers called a_n in this paper). Reprinted in Claude Elwood Shannon: Collected Papers, edited by N. J. A. Sloane and A. D. Wyner, IEEE Press, NY, 1993, pp. 560-570.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Equals (1/2)*A000084 for n >= 2.
Cf. A000055, A000311, A001678, A007827, A363064.
Cf. A000311, labeled hierarchies on n points.
Column 1 of A319254.
Main diagonal of A292085.
Row sums of A292086.

Programs

  • Maple
    Method 1: a := [1,1]; for n from 3 to 30 do L := series( mul( (1-x^k)^(-a[k]),k=1..n-1)/(1-x^n)^b, x,n+1); t1 := coeff(L,x,n); R := series( 1+2*add(a[k]*x^k,k=1..n-1)+2*b*x^n, x, n+1); t2 := coeff(R,x,n); t3 := solve(t1-t2,b); a := [op(a),t3]; od: A000669 := n-> a[n];
Method 2, more efficient: with(numtheory): M := 1001; a := array(0..M); p := array(0..M); a[1] := 1; a[2] := 1; a[3] := 2; p[1] := 1; p[2] := 3; p[3] := 7;
Method 2, cont.: for m from 4 to M do t1 := divisors(m); t3 := 0; for d in t1 minus {m} do t3 := t3+d*a[d]; od: t4 := p[m-1]+2*add(p[k]*a[m-k],k=1..m-2)+t3; a[m] := t4/m; p[m] := t3+t4; od: # A000669 := n-> a[n]; A058757 := n->p[n];
# Method 3:
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(binomial(a(i)+j-1, j)*
       b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n<2, n, b(n, n-1)):
seq(a(n), n=1..40);  # Alois P. Heinz, Jan 28 2016
  • Mathematica
    b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[a[i]+j-1, j]* b[n-i*j, i-1], {j, 0, n/i}]]];
a[n_] := If[n<2, n, b[n, n-1]];
Array[a, 40] (* Jean-François Alcover, Jan 08 2021, after Alois P. Heinz *)
  • PARI
    {a(n) = my(A, X); if( n<2, n>0, X = x + x * O(x^n); A = 1 / (1 - X); for(k=2, n, A /= (1 - X^k)^polcoeff(A, k)); polcoeff(A, n)/2)}; /* Michael Somos, Jul 25 2003 */
  • Sage
    from collections import Counter
def A000669_list(n):
    list = [1] + [0] * (n - 1)
    for i in range(1, n):
        for p in Partitions(i + 1, min_length=2):
            m = Counter(p)
            list[i] += prod(binomial(list[s - 1] + m[s] - 1, m[s]) for s in m)
    return list
print(A000669_list(20)) # M. Eren Kesim, Jun 21 2021

Formula

Product_{k>0} 1/(1-x^k)^a_k = 1+x+2*Sum_{k>1} a_k*x^k.
a(n) ~ c * d^n / n^(3/2), where d = 3.560839309538943329526129172709667..., c = 0.20638144460078903185013578707202765... [Ravelomanana and Thimonier, 2001]. - Vaclav Kotesovec, Aug 25 2014
Consider a nontrivial partition p of n. For each size s of a part occurring in p, compute binomial(a(s)+m-1, m) where m is the multiplicity of s. Take the product of this expression over all s. Take the sum of this new expression over all p to obtain a(n). - Thomas Anton, Nov 22 2018

Extensions

Sequence crossreference fixed by Sean A. Irvine, Sep 15 2009

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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