John Riordan - cp's OEIS frontend

User: John Riordan

John Riordan has authored 3 sequences.

A259859 a(0)=0; thereafter A003470(n-1) + A003470(n) - 1.

0, 1, 3, 10, 38, 177, 999, 6676, 51564, 451585, 4418555, 47746686, 564528978, 7247396065, 100378220943, 1491699317032, 23673159231704, 399553959924801, 7146023007880179, 134997604341408370, 2686037319660797310, 56143557248353416721, 1229914413684635491703

Offset: 0

N. J. A. Sloane, Jul 07 2015, based on a suggestion from John Riordan, Nov 14 1974

nonn

John Riordan and N. J. A. Sloane, Correspondence, 1974

Cf. A003470.

b:= proc(n) b(n):= `if`(n<2, 1, n*b(n-1)-b(n-2)+1+(-1)^n) end:
a:= n-> `if`(n=0, 0, b(n-1)+b(n)-1):
seq(a(n), n=0..22);  # Alois P. Heinz, Jun 17 2021

b[n_] := b[n] = If[n<2, 1, n*b[n-1] - b[n-2] + 1 + (-1)^n];
a[n_] := If[n == 0, 0, b[n-1] + b[n] - 1];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Dec 26 2022, after Alois P. Heinz *)

D-finite with recurrence: (-n+2)*a(n) +(n-1)^2*a(n-1) +2*(-n+2)*a(n-2) +(-n^2+4*n-2)*a(n-3) +(n-1)*a(n-4)=0. - R. J. Mathar, Jul 15 2015

Offset corrected by N. J. A. Sloane, Jun 16 2021

A003470 a(n) = n*a(n-1) - a(n-2) + 1 + (-1)^n.

Original entry on oeis.org

1, 1, 3, 8, 31, 147, 853, 5824, 45741, 405845, 4012711, 43733976, 520795003, 6726601063, 93651619881, 1398047697152, 22275111534553, 377278848390249, 6768744159489931, 128228860181918440, 2557808459478878871, 53585748788874537851, 1176328664895760953853

Offset: 0

N. J. A. Sloane and John Riordan

Row sums of A086764. - Philippe Deléham, Apr 27 2004

a(n+2m) == a(n) (mod m) for all n and m. - Robert Israel, Dec 06 2016

			G.f. = 1 + x + 3*x^2 + 8*x^3 + 31*x^4 + 147*x^5 + 853*x^6 + 5824*x^7 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
John Riordan and N. J. A. Sloane, Correspondence, 1974

Cf. A008279, A072374, A086764, A143409.

f:= gfun:-rectoproc({a(n) -(n-1)*a(n-1)-(n-2)*a(n-2)+a(n-3)-2=0,a(0)=1,a(1)=1,a(2)=3},a(n),remember):
map(f, [$0..30]); # Robert Israel, Dec 06 2016

t = {1, 1}; Do[AppendTo[t, n*t[[-1]] - t[[-2]] + 1 + (-1)^n], {n, 2, 20}] (* T. D. Noe, Oct 07 2013 *)
T[n_, k_] := HypergeometricPFQ[{k+1, k-n},{},-1];
Table[Sum[(-1)^k T[n,k], {k,0,n}], {n,0,22}] (* Peter Luschny, Oct 05 2017 *)

Diagonal sums of reverse of permutation triangle A008279. a(n) = Sum_{k=0..floor(n/2)} (n-k)!/k!. - Paul Barry, May 12 2004
a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*(n-2k)!. - Paul Barry Dec 15 2010
G.f.: 1/(1-x^2-x/(1-x/(1-x^2-2x/(1-2x/(1-x^2-3x/(1-3x/(1-x^2-4x/(1-4x/(1-.... (continued fraction);
G.f.: 1/(1-x-x^2-x^2/(1-3x-x^2-4x^2/(1-5x-x^2-9x^2/(1-7x-x^2-16x^2/(1-... (continued fraction). - Paul Barry, Dec 15 2010
G.f.: hypergeom([1,1],[],x/(1-x^2))/(1-x^2). - Mark van Hoeij, Nov 08 2011
G.f.: 1/Q(0), where Q(k)= 1 - x^2 - x*(k+1)/(1-x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 20 2013
From Robert Israel, Dec 06 2016: (Start)
a(2m) = hypergeom([1,-m,m+1],[],-1).
a(2m+1) = hypergeom([1,-m,m+2],[],-1)*(m+1).
a(2m-1) + a(2m+1) = (2m+1) a(2m). (End)
0 = a(n)*(-a(n+2) - a(n+3)) + a(n+1)*(-2 + a(n+1) - 2*a(n+3) + a(n+4)) + a(n+2)*(-2*a(n+3) + a(n+4)) + a(n+3)*(+2 - a(n+3)) if n >= 0. - Michael Somos, Dec 06 2016
0 = a(n)*(-a(n+2) + a(n+4)) + a(n+1)*(+a(n+1) - a(n+2) - a(n+3) + 3*a(n+4) - a(n+5)) + a(n+2)*(-a(n+3) + a(n+4)) + a(n+3)*(-a(n+4) + a(n+5)) + a(n+4)*(-a(n+4)) if n >= 0. - Michael Somos, Dec 06 2016
a(n) = Sum_{k=0..n} (-1)^k*hypergeom([k+1, k-n], [], -1). - Peter Luschny, Oct 05 2017
D-finite with recurrence: a(n) -n*a(n-1) +(n-2)*a(n-3) -a(n-4)=0. - R. J. Mathar, Apr 29 2020
a(n) ~ n! * (1 + 1/n + 1/(2*n^2) + 2/(3*n^3) + 25/(24*n^4) + 77/(40*n^5) + 2971/(720*n^6) + 6287/(630*n^7) + 1074809/(40320*n^8) + 28160749/(362880*n^9) + ...). - Vaclav Kotesovec, Nov 25 2022

More terms from Gabriel Cunningham (gcasey(AT)mit.edu), Oct 25 2004

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

N. J. A. Sloane and John Riordan

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

			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

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).

N. J. A. Sloane, First 1001 terms of A000669
Mohamed Barakat, Reimer Behrends, Christopher Jefferson, Lukas Kühne, and Martin Leuner, On the generation of rank 3 simple matroids with an application to Terao's freeness conjecture, arXiv:1907.01073 [math.CO], 2019.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Peter J. Cameron, Some treelike objects, Quart. J. Math. Oxford, 38 (1987), 155-183. MR0891613 (89a:05009). See pp. 155, 162, 165, 172. - _N. J. A. Sloane_, Apr 18 2014
Peter J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
Maria Chudnovsky, Jan Goedgebeur, Oliver Schaudt, and Mingxian Zhong, Obstructions for three-coloring graphs without induced paths on six vertices, arXiv preprint arXiv:1504.06979 [math.CO], 2015-2018.
Steven R. Finch, Series-parallel networks
Steven R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
Philippe Flajolet, A Problem in Statistical Classification Theory
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 72
Daniel L. Geisler, Combinatorics of Iterated Functions
A. Genitrini, Full asymptotic expansion for Polya structures, arXiv:1605.00837 [math.CO], May 03 2016, p. 9.
O. Golinelli, Asymptotic behavior of two-terminal series-parallel networks, arXiv:cond-mat/9707023 [cond-mat.stat-mech], 1997.
JiSun Huh and Seonjeong Park, Toric varieties of Schröder type, arXiv:2204.00214 [math.AG], 2022. (Table 1)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 44 [broken link].
V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012. - From _N. J. A. Sloane_, Dec 22 2012
P. A. MacMahon, Yoke-trains and multipartite compositions in connexion with the analytical forms called "trees", Proc. London Math. Soc. 22 (1891), 330-346; reprinted in Coll. Papers I, pp. 600-616. Page 333 gives A000084 = 2*A000669.
Arnau Mir, Francesc Rossello, and Lucia Rotger, Sound Colless-like balance indices for multifurcating trees, arXiv:1805.01329 [q-bio.PE], 2018.
V. Modrak and D. Marton, Development of Metrics and a Complexity Scale for the Topology of Assembly Supply Chains, Entropy 2013, 15, 4285-4299.
J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226.
Vlady Ravelomanana and Loys Thimonier, Asymptotic enumeration of cographs, Electronic Notes in Discrete Mathematics, Volume 7, April 2001, pp. 58-61, Theorem 4.
J. Riordan, The blossoming of Schröder's fourth problem, Acta Math., 137 (1976), 1-16. (page 6)
J. Riordan, Letter to N. J. A. Sloane, Sep. 1970
J. Riordan, Letter to N. J. A. Sloane, Nov 10 1970
Audace Amen Vioutou Dossou-Olory, and Stephan Wagner, Inducibility of Topological Trees, arXiv:1802.06696 [math.CO], 2018.
Audace Amen Vioutou Dossou-Olory, The topological trees with extremal Matula numbers, arXiv:1806.03995 [math.CO], 2018.
Wei Wang and Ximei Huang, Almost all cographs have a cospectral mate, arXiv:2507.16730 [math.CO], 2025. See pp. 6, 8.
Eric Weisstein's World of Mathematics, Series-Parallel Network
Index entries for sequences related to rooted trees
Index entries for sequences mentioned in Moon (1987)
Index entries for sequences related to trees

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.

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

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 *)

{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 */

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

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

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

cp's OEIS Frontend

User: John Riordan

A259859 a(0)=0; thereafter A003470(n-1) + A003470(n) - 1.

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A003470 a(n) = n*a(n-1) - a(n-2) + 1 + (-1)^n.

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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.

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