A000084 -id:A000084 - cp's OEIS frontend

A000137 Series-parallel numbers.

1, 2, 6, 18, 58, 186, 614, 2034, 6818, 22970, 77858, 264970, 905294, 3102434, 10661370, 36722642, 126752218, 438294018, 1518032598, 5265341314, 18286911130, 63586988434, 221342104842, 771235606050, 2689688538646, 9388096331642

Offset: 1

N. J. A. Sloane

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 142.
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).

Sean A. Irvine, Table of n, a(n) for n = 1..1000

Cf. A036654, A036655, A058756, A000084.

n = 30; s = 1/(1 - x) + O[x]^(n + 1); Do[s = s/(1 - x^k)^Coefficient[s, x^k] + O[x]^(n + 1), {k, 2, n}] ; S = s - 1; CoefficientList[(1 + S)/(1 - S) + O[x]^n, x] (* Jean-François Alcover, Feb 09 2016 *)

G.f.: x(1+S)/(1-S), where S = g.f. for A000084.

A001572 Related to series-parallel networks.

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 17, 41, 127, 365, 1119, 3413, 10685, 33561, 106827, 342129, 1104347, 3584649, 11701369, 38374065, 126395259, 417908329, 1386618307, 4615388353, 15407188529, 51569669429, 173033992311, 581905285089, 1961034571967

Offset: 0

N. J. A. Sloane

From Gary W. Adamson, Sep 27 2008: (Start)
Starting (1, 1, 1, 3, 5, 17, ...) = the INVERTi transform of A000084: (1, 2, 4, 10, 24, 66, ...).
Equals left border of triangle A144962. (End)

J. Riordan and C. E. Shannon, The number of two-terminal series-parallel networks, J. Math. Phys., 21 (1942), 83-93. 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).

J. Riordan and C. E. Shannon, The number of two-terminal series-parallel networks (annotated scanned copy)

Cf. A000084, A144962. - Gary W. Adamson, Sep 27 2008

max = 29;(* b = A000669 *) b[1] = 1; b[n_] := Module[{s}, s = Series[1/(1 - x), {x, 0, n}]; Do[s = Series[s/(1 - x^k)^Coefficient[s, x^k], {x, 0, n}], {k, 2, n}]; Coefficient[s, x^n]/2]; gf = 2 - Product[(1 - x^n)^b[n], {n, 1, max}] + O[x]^max; CoefficientList[gf, x] (* Jean-François Alcover, Oct 23 2016 *)

G.f.: 1 - Sum_{k>=1} a(k)*x^k = Product_{n>=1} (1-x^n)^A000669(n).

A001677 Number of series-parallel networks with n edges.

Original entry on oeis.org

1, 2, 3, 6, 12, 26, 59, 146, 368, 976, 2667, 7482, 21440, 62622, 185637, 557680, 1694256, 5198142, 16086486, 50165218, 157510504, 497607008, 1580800091, 5047337994, 16190223624, 52153429218, 168657986843, 547389492416

Offset: 2

N. J. A. Sloane

			a(5) = 24 - (1/2)*(1*10+2*4+4*2+10*1) = 6.

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).
B. D. H. Tellegen, Geometrical configurations and duality of electrical networks, Philips Technical Review, 5 (1940), 324-330.

T. D. Noe, Table of n, a(n) for n=2..500
R. M. Foster, The number of series-parallel networks, Proc. Intern. Congr. Math., Vol. 1, 1950, p. 646.

Cf. A058642, A058668.

m = 29; ClearAll[a, b, s]; a[1] = 1; a[2] = 2; a[3] = 4; b[1] = 1; b[n_ /; n >= 2] = a[n]/2; ex = Product[ 1/(1-x^k)^b[k], {k, 1, m}] - 1 - Sum[ a[k]*x^k, {k, 1, m}]; coes = CoefficientList[ Series[ ex, {x, 0, m}], x]; sol = Solve[ Thread[ coes == 0]][[1]]; Do[ s[k] = a[k] /. sol, {k, 1, m}]; a[2] = 1; a[3] = 2; a[n_] := s[n] - (1/2)*Sum[ s[i]*s[n-i], {i, 1, n-1}] - If[ OddQ[n], 0, s[n/2]/2]; Table[ a[n], {n, 2, m}] (* Jean-François Alcover, Feb 24 2012 *)

a(n) = s(n) - (1/2)*Sum_{i=1..n-1} s(i)*s(n-i) - (1/2)*s(n/2), where s() = A000084 and the last term is omitted if n is odd.

More terms from David W. Wilson, Sep 20 2000

A036655 Numbers d_n used in recurrence for series-parallel numbers.

Original entry on oeis.org

1, 2, 2, 6, 2, 18, 2, 46, 14, 126, 2, 454, 2, 1266, 134, 4222, 2, 14202, 2, 46406, 1274, 155502, 2, 531790, 122, 1792810, 13802, 6126334, 2, 21032794, 2, 72121854, 155510, 248396798, 1382, 857944150, 2, 2964896882, 1792818, 10269600622

Offset: 0

N. J. A. Sloane

J. Riordan, An Introduction to Combinatorial Analysis, foot of p. 141.

G.f.: S'*(1-S)/(1+S), where S = g.f. for A000084.

A058386 Essentially series series-parallel networks with n unlabeled edges, multiple edges not allowed.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 9, 20, 47, 112, 274, 678, 1709, 4346, 11176, 28966, 75656, 198814, 525496, 1395758, 3723986, 9975314, 26817655, 72332320, 195679137, 530814386, 1443556739, 3934880554, 10748839215, 29420919456, 80678144437, 221618678694

Offset: 0

N. J. A. Sloane, Dec 20 2000

Steven R. Finch, Series-parallel networks.
Steven R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
John W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226 (the sequence r_n).
Index entries for sequences mentioned in Moon (1987)

Cf. A058379, A058385, A058387.

(* f = g.f. of A058385 *) max = 31; f[x_] := Sum[b[n]*x^n, {n, 0, max}]; b[0] = 0; b[1] = 1; b[2] = 0; b[3] = 1; coef = CoefficientList[ Series[1 - x + x^2 + 2*f[x] - Product[(1 - x^j)^(-b[j]), {j, 1, max}], {x, 0, max}], x][[ 5 ;; All]]; g[x_] := Sum[a[n]*x^n, {n, 0, max}]; a[0] = a[1] = 0; a[2] = a[3] = 1; coeg = CoefficientList[ Series[g[x] - f[x] + x - x^2, {x, 0, max}], x][[ 5 ;; All]]; solf = SolveAlways[ Thread[coef == 0], x] ; solg = SolveAlways[ Thread[coeg == 0] /. solf[[1]], x]; Table[a[n], {n, 0, max}] /. solg[[1]] (* Jean-François Alcover, Jul 18 2012 *)
terms = 32; (* f = g.f. of A058385 *) f[] = 0; Do[f[x] = (1/2)*(-1 + x - x^2 + Product[(1 - x^j)^(-Ceiling[Coefficient[f[x], x, j]]), {j, 1,  terms}]) + O[x]^ terms // Normal, 4*terms]; A[x_] = f[x] - x + x^2 + O[x]^terms; CoefficientList[A[x], x] (* Jean-François Alcover, Jan 10 2018 *)

G.f. satisfies A(x) = A058385(x) - x + x^2.

A058964 Decimal expansion of series-parallel constant.

Original entry on oeis.org

2, 8, 0, 8, 3, 2, 6, 6, 6, 9, 8, 4, 2, 0, 0, 3, 5, 5, 3, 9, 3, 2

Offset: 0

N. J. A. Sloane, E. M. Rains, Jan 14 2001

			0.2808326669842003553932...

J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226.
J. Riordan and C. E. Shannon, The number of two-terminal series-parallel networks, J. Math. Phys., 21 (1942), 83-93. Reprinted in Claude Elwood Shannon: Collected Papers, edited by N. J. A. Sloane and A. D. Wyner, IEEE Press, NY, 1993, pp. 560-570.

Steven R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
O. Golinelli, Asymptotic behavior of two-terminal series-parallel networks, arXiv:cond-mat/9707023 [cond-mat.stat-mech], 1997.

See A058965 for continued fraction expansion. Cf. A000084, A000669.

This number, c, is defined by Product_{n=1..inf} (1-c^n)^(-A000669[n]) = 2.

A058965 Continued fraction expansion of series-parallel constant.

Original entry on oeis.org

0, 3, 1, 1, 3, 1, 1, 1, 1, 3, 1, 3, 12, 1, 8, 8, 1, 7, 6, 1, 5, 2, 1, 1, 4, 1, 3, 2, 36, 1, 10, 6, 1, 2

Offset: 0

N. J. A. Sloane, E. M. Rains, Jan 14 2001

			Constant is 0.2808326669842003553932...

J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226.
J. Riordan and C. E. Shannon, The number of two-terminal series-parallel networks, J. Math. Phys., 21 (1942), 83-93. Reprinted in Claude Elwood Shannon: Collected Papers, edited by N. J. A. Sloane and A. D. Wyner, IEEE Press, NY, 1993, pp. 560-570.

Steven R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
O. Golinelli, Asymptotic behavior of two-terminal series-parallel networks, arXiv:cond-mat/9707023 [cond-mat.stat-mech], 1997.

See A058964 for decimal expansion. Cf. A000084, A000669.

This number, c, is defined by Product_{n=1..inf} (1-c^n)^(-A000669[n]) = 2.

A000163 Series-parallel numbers.

Original entry on oeis.org

2, 8, 34, 136, 538, 2080, 7970, 30224, 113874, 426520, 1590338, 5906640, 21866794, 80725280, 297292610, 1092539736, 4007553922, 14675748416, 53663197698, 195962433328, 714734393378, 2603994002280, 9477643739746, 34463730424720

Offset: 2

N. J. A. Sloane

nonn

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 142.
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).

Sean A. Irvine, Table of n, a(n) for n = 2..500

n = 25; s = 1/(1 - x) + O[x]^(n + 1); Do[s = s/(1 - x^k)^Coefficient[s, x^k] + O[x]^(n + 1), {k, 2, n}] ; S = s - 1; CoefficientList[2*(1 + S)/(1 - S)^3 + O[x]^n, x] (* Jean-François Alcover, Feb 09 2016 *)

G.f.: 2*(1+S)/(1-S)^3, where S = g.f. for A000084. [Sean A. Irvine, Oct 14 2009]

More terms from Sean A. Irvine, Oct 14 2009

A046906 Number of connected irreducible posets with n labeled points.

Original entry on oeis.org

1, 1, 0, 0, 24, 1080, 52440, 3281880, 277953144, 32418855000, 5239070305080, 1173944480658840, 363936227764858584, 155521768202208047640, 91218870039317505477720, 73113879800794757415243480, 79743817918540500914682249144, 117883366412734188786535902826200, 235329353612778837110901775412557560

Offset: 0

John A. Wright

nonn

J. A. Wright, There are 718 6-point topologies, quasi-orderings and transgraphs, Notices Amer. Math. Soc., 17 (1970), p. 646, Abstract #70T-A106.

S. R. Finch, Series-parallel networks
S. R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
J. A. Wright, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences
Index entries for sequences related to posets

A003431 gives isomorphism classes of these posets.

nn = 18; A[x_] := Total[Cases[Import["https://oeis.org/A001035/b001035.txt",
      "Table"], {, }][[All, 2]]*Table[x^(i - 1)/(i - 1)!, {i, 1, 19}]];
Range[0, nn]! CoefficientList[ Series[(1 + Log[A[x]]) - A[ x] (1 - 1/A[x])^2 , {x, 0, nn}], x] (* Geoffrey Critzer, Jul 09 2022 *)

From Geoffrey Critzer, Jul 09 2022: (Start)
E.g.f.: 1 + log(A(x)) - A(x)(1-1/A(x))^2  where A(x) is the e.g.f. for A001035.
a(n) = A001927(n) - Sum_{k>=2} A354615(n,k). (End)

a(8)-a(18) from Geoffrey Critzer, Jul 09 2022

a(0) changed to 1 by Geoffrey Critzer, Jul 10 2022

A352213 Largest number of maximal cographical node-induced subgraphs of an n-node graph.

Original entry on oeis.org

1, 1, 1, 4, 10, 12, 23, 38, 64

Offset: 1

Pontus von Brömssen, Mar 08 2022

This sequence is log-superadditive, i.e., a(m+n) >= a(m)*a(n). By Fekete's subadditive lemma, it follows that the limit of a(n)^(1/n) exists and equals the supremum of a(n)^(1/n).

			All graphs with at most three nodes are cographs, so a(n) = 1 for n <= 3 and any graph is optimal.
All optimal graphs (i.e., graphs that have n nodes and a(n) maximal cographical subgraphs) for 4 <= n <= 9 are listed below. Since a graph is a cograph if and only if its complement is a cograph, the optimal graphs come in complementary pairs.
  n = 4: the path of length 3 (self-complementary);
  n = 5: the 5-cycle (self-complementary);
  n = 6: the Hajós graph (also known as a Sierpiński sieve graph) and its complement;
  n = 7: the elongated triangular pyramid and its complement;
  n = 8: the Möbius ladder and its complement (the antiprism graph);
  n = 9: the pentagonal bipyramid with an additional path of length 3 between the two apex nodes (self-complementary).

Cf. A000084, A000669.

For a list of related sequences, see cross-references in A342211.

a(m+n) >= a(m)*a(n).

Limit_{n->oo} a(n)^(1/n) >= 64^(1/9) = 1.58740... .

cp's OEIS Frontend

A000137 Series-parallel numbers.

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A001572 Related to series-parallel networks.

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A001677 Number of series-parallel networks with n edges.

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A036655 Numbers d_n used in recurrence for series-parallel numbers.

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A058386 Essentially series series-parallel networks with n unlabeled edges, multiple edges not allowed.

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A058964 Decimal expansion of series-parallel constant.

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A058965 Continued fraction expansion of series-parallel constant.

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A000163 Series-parallel numbers.

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A046906 Number of connected irreducible posets with n labeled points.

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