A275166 -id:A275166 - cp's OEIS frontend

A216785 Number of unlabeled graphs on n nodes that have exactly two non-isomorphic components.

0, 0, 1, 2, 8, 28, 145, 1022, 12320, 274785, 12007355, 1019030127, 165091859656, 50502058491413, 29054157815353374, 31426486309136268658, 64001015806929213894372, 245935864212056913811498454, 1787577725208700551275529005084, 24637809253253259917745389824933448

Offset: 1

Geoffrey Critzer, Oct 15 2012

nonn

Stated more precisely: Number of unlabeled graphs on n nodes that have exactly two connected components and these components are not isomorphic (and nonempty).

			a(4)=2 = 1*2 where 1*2=A001349(1)*A001349(3) counts graphs with a component of 1 node and a component with 3 nodes. There is no contribution with a component of 2 nodes and another component of 2 nodes because A001349(2)=1 means these components would be isomorphic. - _R. J. Mathar_, Jul 18 2016
a(5)=8 = 1*6 + 1*2 where 1*6=A001349(1)*A001349(4) counts graphs with a component of 1 node and a component with 4 nodes, and where 1*2 = A001349(2)*A001349(3) counts graphs with a component of 2 nodes and a component of 3 nodes. - _R. J. Mathar_, Jul 18 2016

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973, page 48.

David Broadhurst, Table of n, a(n) for n = 1..76

Cf. A058915, A001349, A217955, A275165, A275166 (allows an empty component), A274934 (allows isomorphic components).

Needs["Combinatorica`"]; max=25; A000088=Table[NumberOfGraphs[n], {n, 0, max}]; f[x_]=1-Product[1/(1-x^k)^a[k], {k, 1, max}]; a[0]=a[1]=a[2]=1; coes=CoefficientList[Series[f[x], {x, 0, max}], x]; sol=First[Solve[Thread[Rest[coes+A000088]==0]]]; cg=Table[a[n], {n, 1, max}]/.sol; Take[CoefficientList[CycleIndex[AlternatingGroup[2], s]-CycleIndex[SymmetricGroup[2], s]/.Table[s[j]->Table[Sum[cg[[i]] x^(k*i), {i, 1, max}], {k, 1, max}][[j]], {j, 1, 3}], x], {4, max}]  (* after code given by Jean-François Alcover in A001349 *)

{c=[1, 1, 2, 6, 21, 112, 853, 11117, 261080, 11716571, 1006700565, 164059830476, 50335907869219, 29003487462848061, 31397381142761241960, 63969560113225176176277, 245871831682084026519528568, 1787331725248899088890200576580, 24636021429399867655322650759681644];}
for(n=0,19,print([n,sum(j=1,(n-1)\2,c[j]*c[n-j])+if(n%2==0,c[n/2]*(c[n/2]-1)/2)])); /* David Broadhurst, Jul 18 2016 */

from functools import lru_cache
from itertools import combinations
from fractions import Fraction
from math import prod, gcd, factorial, comb
from sympy import mobius, divisors
from sympy.utilities.iterables import partitions
def A216785(n):
    @lru_cache(maxsize=None)
    def b(n): return int(sum(Fraction(1<>1)*r+(q*r*(r-1)>>1) for q, r in p.items()),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n)))
    @lru_cache(maxsize=None)
    def c(n): return n*b(n)-sum(c(k)*b(n-k) for k in range(1,n))
    @lru_cache(maxsize=None)
    def d(n): return sum(mobius(n//d)*c(d) for d in divisors(n,generator=True))//n
    return sum(d(i)*d(n-i) for i in range(1,n+1>>1)) + (0 if n&1 else comb(d(n>>1),2)) # Chai Wah Wu, Jul 03 2024

O.g.f.: A(x)^2/2 - A(x^2)/2 where A(x) is the o.g.f. for A001349 after setting A001349(0)=0.

Two zeros prepended (offset changed), formula updated, and entries corrected by R. J. Mathar, N. J. A. Sloane, Jul 18 2016. (Thanks to Allan C. Wechsler for pointing out that all the entries above a(19) were wrong.)

A274934 Number of unlabeled graphs with n nodes that have two components, neither of which is the empty graph.

Original entry on oeis.org

0, 0, 1, 1, 3, 8, 30, 145, 1028, 12320, 274806, 12007355, 1019030239, 165091859656, 50502058492266, 29054157815353374, 31426486309136279775, 64001015806929213894372, 245935864212056913811759534, 1787577725208700551275529005084

Offset: 0

R. J. Mathar and N. J. A. Sloane, Jul 18 2016

nonn

			a(6) = A216785(6)+2 =30 where the two additional graphs have two equal components (of which there are A001349(3)=2 choices).

Alois P. Heinz, Table of n, a(n) for n = 0..75
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017) Table 81.

Cf. A001349, A216785 (non-isomorphic components), A275165, A275166, column 2 of A201922.

terms = 20;
mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2]];
a88[n_] := Module[{s = 0}, Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!];
A[x_] = Join[{1}, EULERi[Array[a88, terms]]].x^Range[0, terms] - 1;
CoefficientList[(A[x]^2 + A[x^2])/2 + O[x]^terms, x] (* Jean-François Alcover, Sep 28 2018, after Andrew Howroyd in A001349 *)

G.f.: [A(x)^2 + A(x^2)]/2 where A(x) is the o.g.f. for A001349 without the initial constant 1.

a(n) = A201922(n,2). - R. J. Mathar, Jul 20 2016

A275165 Number of n-node graphs with two connected components.

Original entry on oeis.org

1, 1, 2, 3, 9, 29, 142, 998, 12145, 273400, 11991377, 1018707920, 165078860715, 50500999728875, 29053989521340327, 31426435300576595334, 64000986599534312456052, 245935832697890955733422940, 1787577661113111145804012336114, 24637809007125076355873926288686728

Offset: 0

R. J. Mathar, Jul 18 2016

nonn

"Component" means there are no edges from a node of one component to any node of the other component.
Each of the 2 components may be the empty graph with 0 nodes. That means the graph has only one "visible" component in these cases.
Each of the 2 components must be a connected graph (see A001349). (The empty graph has all properties and is a connected graph.)
The graphs of the components may be the same (=isomorphic).

			a(4)=9 = 1*6 + 1*2 + 1*1 where 1*6=A001349(0)*A001349(4) counts graphs with an empty component and a component with 4 nodes, where 1*2 = A001349(1)*A001349(3) counts graphs with a component of 1 node and a component of 3 nodes, and where 1*1 = A001349(2)*A001349(2) counts graph with a component of 2 nodes and another component of 2 nodes (both components the same in that case).

Alois P. Heinz, Table of n, a(n) for n = 0..75

Cf. A216785, A001349, A275166, A274934 (no empty components).

terms = 20;
mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++,c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] +
   Total[Quotient[v, 2]];
a88[n_] := Module[{s = 0}, Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!];
A[x_] = Join[{1}, EULERi[Array[a88, terms]]].(x^Range[0, terms]);
CoefficientList[(A[x]^2 + A[x^2])/2 + O[x]^terms, x] (* Jean-François Alcover, May 28 2019, after Andrew Howroyd in A001349 *)

from functools import lru_cache
from itertools import combinations
from fractions import Fraction
from math import prod, gcd, factorial, comb
from sympy import mobius, divisors
from sympy.utilities.iterables import partitions
def A275165(n):
    @lru_cache(maxsize=None)
    def b(n): return int(sum(Fraction(1<>1)*r+(q*r*(r-1)>>1) for q, r in p.items()),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n)))
    @lru_cache(maxsize=None)
    def c(n): return n*b(n)-sum(c(k)*b(n-k) for k in range(1,n))
    @lru_cache(maxsize=None)
    def d(n): return sum(mobius(n//d)*c(d) for d in divisors(n,generator=True))//n if n else 1
    return sum(d(i)*d(n-i) for i in range(n+1>>1)) + (0 if n&1 else comb(d(n>>1)+1,2)) # Chai Wah Wu, Jul 03 2024

G.f.: [A(x)^2 + A(x^2)]/2 where A(x) is the o.g.f. for A001349.

a(n) = A275166(n) if n odd.

A000150 Number of dissections of an n-gon, rooted at an exterior edge, asymmetric with respect to that edge.

Original entry on oeis.org

0, 0, 1, 2, 7, 20, 66, 212, 715, 2424, 8398, 29372, 104006, 371384, 1337220, 4847208, 17678835, 64821680, 238819350, 883629164, 3282060210, 12233125112, 45741281820, 171529777432, 644952073662, 2430973096720, 9183676536076

Offset: 0

N. J. A. Sloane

Number of Dyck paths of length 2n having an odd number of peaks at even height. Example: a(3)=2 because we have UDU(UD)D and U(UD)DUD, where U=(1,1), D=(1,-1) and the peaks at even height are shown between parentheses. - Emeric Deutsch, Nov 13 2004
For n>=1, a(n) is the number of unordered binary trees with n internal nodes in which the left subtree is distinct from the right subtree. - Geoffrey Critzer, Feb 21 2013
Assuming offset -1 this is an analog of A275166: pairs of distinct Catalan numbers with index sum n. - R. J. Mathar, Jul 19 2016

S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751
R. K. Guy, "Dissecting a polygon into triangles," Bull. Malayan Math. Soc., Vol. 5, pp. 57-60, 1958.
R. K. Guy, Dissecting a polygon into triangles, Research Paper #9, Math. Dept., Univ. Calgary, 1967.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 78, (3.5.26).
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).
P. K. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974.

T. D. Noe, Table of n, a(n) for n=0..200
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]
R. K. Guy, Dissecting a polygon into triangles, Research Paper #9, Math. Dept., Univ. Calgary, 1967. [Annotated scanned copy]
F. Harary and E. M. Palmer, On acyclic simplicial complexes, Mathematika 15 1968 115-122.
Krishna Menon and Anurag Singh, Grassmannian permutations avoiding identity, arXiv:2212.13794 [math.CO], 2022.
P. J. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974. [Scanned annotated and corrected copy]
Index entries for sequences related to Lyndon words

a(n) = T(2n+2, n), array T as in A051168, a count of Lyndon words.
Cf. A051168, A005430.
Cf. A007595.
A diagonal of the square array described in A051168.

nn=20;CoefficientList[Series[x/2(((1-(1-4x)^(1/2))/(2x))^2-(1-(1-4x^2)^(1/2))/(2x^2)),{x,0,nn}],x]  (* Geoffrey Critzer, Feb 21 2013 *)

Let c(x) = (1-sqrt(1-4*x))/(2*x) = g.f. for Catalan numbers (A000108), let d(x) = 1+x*c(x^2). Then g.f. is (c(x)-d(x))/2.
G.f.: (sqrt(1-4*z^2) - sqrt(1-4*z) - 2*z)/(4*z). - Emeric Deutsch, Nov 13 2004
With c(x) defined as above: g.f. = x*(c(x)^2/2 - c(x^2)/2). - Geoffrey Critzer, Feb 21 2013
a(n) = ( 2^(n-3)/sqrt(Pi) ) * ( 4*2^n*GAMMA(n+1/2)/GAMMA(n+2) + ((-1)^n - 1)*GAMMA(n/2)/GAMMA(n/2 + 3/2) ) for n>0. - Mark van Hoeij, Nov 11 2009
a(n) ~ 2^(2*n-1) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 10 2014
a(2n) = A000108(2n) / 2; a(2n+1) = ( A000108(2n+1) - A000108(n) ) / 2. - John Bodeen, Jun 24 2015
D-finite with recurrence +n*(n+1)*(n-2)^2*a(n) -2*n*(2*n-5)*(n-1)^2*a(n-1) -4*n*(n-2)^3*a(n-2) +8*(2*n-5)*(n-3)*(n-1)^2*a(n-3)=0. - R. J. Mathar, Oct 28 2021

Additional comments from Clark Kimberling

A274937 Number of unlabeled forests on n nodes that have exactly two nonempty components.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 6, 11, 23, 46, 99, 216, 488, 1121, 2644, 6334, 15437, 38132, 95368, 241029, 614968, 1582030, 4100157, 10697038, 28075303, 74086468, 196470902, 523383136, 1400051585, 3759508536, 10131097618, 27391132238, 74283552343, 202030012554, 550934060120, 1506161266348

Offset: 0

N. J. A. Sloane, Jul 19 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000055, A274935, A274936, A274938. [A274935, A274936, A274937, A274938] are analogs for forests of [A275165, A275166, A216785, A274934] for graphs.

b:= proc(n) option remember; `if`(n<2, n, (add(add(d*
      b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
    end:
g:= proc(n) option remember; `if`(n=0, 1, b(n)-add(b(j)*
      b(n-j), j=0..n/2)+`if`(n::odd, 0, (t->t*(t+1)/2)(b(n/2))))
    end:
a:= proc(n) option remember; add(g(j)*g(n-j), j=1..n/2)-
      `if`(n::odd, 0, (t-> t*(t-1)/2)(g(n/2)))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 20 2016

b[n_] := b[n] = If[n<2, n, Sum[DivisorSum[j, #*b[#]&]*b[n-j], {j, 1, n-1}]/ (n-1)];
g[n_] := g[n] = If[n==0, 1, b[n]-Sum[b[j]*b[n-j], {j, 0, n/2}] + If[OddQ[n], 0, Function[t, t*(t+1)/2][b[n/2]]]];
a[n_] := a[n] = Sum[g[j]*g[n-j], {j, 1, n/2}] - If[OddQ[n], 0, Function[t, t*(t-1)/2][g[n/2]]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 14 2017, after Alois P. Heinz *)

G.f.: [A(x)^2 + A(x^2)]/2 where A(x) is the o.g.f. for A000055 without the initial constant 1.

a(n) = A095133(n,2). - R. J. Mathar, Jul 20 2016

A274935 Number of n-node unlabeled forests with two connected components.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 12, 22, 46, 93, 205, 451, 1039, 2422, 5803, 14075, 34757, 86761, 219235, 558984, 1438033, 3726535, 9723913, 25525112, 67375200, 178723358, 476264352, 1274448596, 3423494617, 9229075121, 24961969420, 67721961268, 184255962564, 502658875034, 1374713691841, 3768527610094, 10353602341313

Offset: 0

N. J. A. Sloane, Jul 19 2016

nonn

One of the components may be empty (the null graph): a(n) = A000055(n) + A274937(n). - R. J. Mathar, Aug 15 2017

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000055, A274936, A274937, A274938. [A274935, A274936, A274937, A274938] are analogs for forests of [A275165, A275166, A216785, A274934] for graphs.

with(numtheory):
b:= proc(n) option remember; `if`(n<2, n, (add(add(d*
      b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
    end:
g:= proc(n) option remember; `if`(n=0, 1, b(n)-add(b(j)*
      b(n-j), j=0..n/2)+`if`(n::odd, 0, (t->t*(t+1)/2)(b(n/2))))
    end:
a:= proc(n) option remember; add(g(j)*g(n-j), j=0..n/2)-
      `if`(n::odd, 0, (t-> t*(t-1)/2)(g(n/2)))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 20 2016

b[n_] := b[n] = If[n<2, n, Sum[DivisorSum[j, #*b[#]&]*b[n-j], {j, 1, n-1}]/ (n-1)];
g[n_] := g[n] = If[n==0, 1, b[n]-Sum[b[j]*b[n-j], {j, 0, n/2}]+If[OddQ[n], 0, Function[t, t*(t+1)/2][b[n/2]]]];
a[n_] := a[n] = Sum[g[j]*g[n-j], {j, 0, n/2}]-If[OddQ[n], 0, Function[t, t*(t-1)/2][g[n/2]]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 15 2017, after Alois P. Heinz *)

G.f.: [A(x)^2 + A(x^2)]/2 where A(x) is the o.g.f. for A000055.

A274936 Number of n-node unlabeled forests that have 2 non-isomorphic components.

Original entry on oeis.org

0, 1, 1, 2, 3, 6, 11, 22, 44, 93, 202, 451, 1033, 2422, 5792, 14075, 34734, 86761, 219188, 558984, 1437927, 3726535, 9723678, 25525112, 67374649, 178723358, 476263051, 1274448596, 3423491458, 9229075121, 24961961679, 67721961268, 184255943244, 502658875034, 1374713643212

Offset: 0

N. J. A. Sloane, Jul 19 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000055, A274935-A274938. [A274935, A274936, A274937, A274938] are analogs for forests of [A275165, A275166, A216785, A274934] for graphs.

with(numtheory):
b:= proc(n) option remember; `if`(n<2, n, (add(add(d*
      b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
    end:
g:= proc(n) option remember; `if`(n=0, 1, b(n)-add(b(j)*
      b(n-j), j=0..n/2)+`if`(n::odd, 0, (t->t*(t+1)/2)(b(n/2))))
    end:
a:= proc(n) option remember; add(g(j)*g(n-j), j=0..n/2)-
      `if`(n::odd, 0, (t-> t*(t+1)/2)(g(n/2)))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 20 2016

b[n_] := b[n] = If[n<2, n, Sum[DivisorSum[j, #*b[#]&]*b[n-j], {j, 1, n-1}]/(n-1)];
g[n_] := g[n] = If[n==0, 1, b[n]-Sum[b[j]*b[n-j], {j, 0, n/2}]+If[OddQ[n], 0, Function[t, t*(t+1)/2][b[n/2]]]];
a[n_] := a[n] = Sum[g[j]*g[n-j], {j, 0, n/2}]-If[OddQ[n], 0, Function[t, t*(t+1)/2][g[n/2]]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 15 2017, after Alois P. Heinz *)

G.f.: [A(x)^2 - A(x^2)]/2 where A(x) is the o.g.f. for A000055.

a(2n+1) = A274935(2n+1). a(2n) = A274935(2n)-A000055(n). - R. J. Mathar, Jul 20 2016

A274938 Number of unlabeled forests with n nodes that have two components, neither of which is the empty graph.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 5, 11, 21, 46, 96, 216, 482, 1121, 2633, 6334, 15414, 38132, 95321, 241029, 614862, 1582030, 4099922, 10697038, 28074752, 74086468, 196469601, 523383136, 1400048426, 3759508536, 10131089877, 27391132238, 74283533023, 202030012554, 550934011491, 1506161266348

Offset: 0

N. J. A. Sloane, Jul 19 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000055, A274935-A274937. [A274935, A274936, A274937, A274938] are analogs for forests of [A275165, A275166, A216785, A274934] for graphs.

with(numtheory):
b:= proc(n) option remember; `if`(n<2, n, (add(add(d*
      b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
    end:
g:= proc(n) option remember; `if`(n=0, 1, b(n)-add(b(j)*
      b(n-j), j=0..n/2)+`if`(n::odd, 0, (t->t*(t+1)/2)(b(n/2))))
    end:
a:= proc(n) option remember; add(g(j)*g(n-j), j=1..n/2)-
      `if`(n::odd or n=0, 0, (t-> t*(t+1)/2)(g(n/2)))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 20 2016

b[n_] := b[n] = If[n<2, n, Sum[DivisorSum[j, #*b[#]&]*b[n-j], {j, 1, n-1}]/(n-1)];
g[n_] := g[n] = If[n==0, 1, b[n]-Sum[b[j]*b[n-j], {j, 0, n/2}]+If[OddQ[n], 0, Function[t, t*(t+1)/2][b[n/2]]]];
a[n_] := a[n] = Sum[g[j]*g[n-j], {j, 1, n/2}]-If[OddQ[n] || n==0, 0, Function[t, t*(t+1)/2][g[n/2]]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 15 2017, after Alois P. Heinz *)

G.f.: [A(x)^2 - A(x^2)]/2 where A(x) is the o.g.f. for A000055 without the initial constant 1.

a(2n+1) = A274937(2n+1). a(2n) = A274937(2n)-A000055(n). - R. J. Mathar, Jul 20 2016

A275208 Expansion of (A(x)^2-A(x^2))/2 where A(x) = A001006(x).

Original entry on oeis.org

0, 1, 2, 6, 14, 38, 96, 256, 672, 1805, 4846, 13162, 35874, 98469, 271384, 751656, 2089640, 5831451, 16325950, 45847770, 129106738, 364498596, 1031480792, 2925337352, 8313200232, 23668977163, 67507731786, 192859753310, 551821286374, 1581188102590

Offset: 0

R. J. Mathar, Jul 19 2016

nonn

Analog of A275166 with Motzkin numbers replacing connected graph counts.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A275207, A026940.

b:= proc(n) option remember; `if`(n<2, 1,
      ((3*(n-1))*b(n-2)+(1+2*n)*b(n-1))/(n+2))
    end:
a:= proc(n) option remember; add(b(j)*b(n-j), j=0..n/2)-
      `if`(n::odd, 0, (t-> t*(t+1)/2)(b(n/2)))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 19 2016

b[n_] := b[n] = If[n<2, 1, ((3*(n-1))*b[n-2] + (1+2*n)*b[n-1])/(n+2)];
a[n_] := a[n] = Sum[b[j]*b[n-j], {j, 0, n/2}] - If[OddQ[n], 0, Function[t, t*(t + 1)/2][b[n/2]]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 16 2017, after Alois P. Heinz *)

a(2n+1) = A275207(2n+1).

cp's OEIS Frontend

A216785 Number of unlabeled graphs on n nodes that have exactly two non-isomorphic components.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A274934 Number of unlabeled graphs with n nodes that have two components, neither of which is the empty graph.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A275165 Number of n-node graphs with two connected components.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A000150 Number of dissections of an n-gon, rooted at an exterior edge, asymmetric with respect to that edge.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A274937 Number of unlabeled forests on n nodes that have exactly two nonempty components.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A274935 Number of n-node unlabeled forests with two connected components.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A274936 Number of n-node unlabeled forests that have 2 non-isomorphic components.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica