A216785 -id:A216785 - cp's OEIS frontend

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

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.

A275166 Number of n-node graphs that have 2 non-isomorphic components.

Original entry on oeis.org

0, 1, 1, 3, 8, 29, 140, 998, 12139, 273400, 11991356, 1018707920, 165078860603, 50500999728875, 29053989521339474, 31426435300576595334, 64000986599534312444935, 245935832697890955733422940, 1787577661113111145804012075034, 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 2 components must not be the same (not be isomorphic).

			a(4)=8 = 1*6 + 1*2 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. There is no contribution from a component of 2 nodes and another component of 2 nodes (both components were isomorphic in that case).

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

Cf. A216785, A001349, A275165.

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];
(A[x]^2 - A[x^2])/2 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Jan 31 2020, 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 A275166(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),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) = A275165(n) if n odd.

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

A217955 Triangular array read by rows. T(n,k) is the number of unlabeled graphs on n nodes that have exactly k distinct components (n >= 1).

Original entry on oeis.org

1, 1, 2, 1, 6, 2, 21, 8, 112, 28, 2, 853, 145, 7, 11117, 1022, 34, 261080, 12320, 181, 1, 11716571, 274785, 1266, 12, 1006700565, 12007355, 14106, 63, 164059830476, 1019030127, 293756, 407, 50335907869219, 165091859656, 12362198, 3023, 6, 29003487462848061, 50502058491413, 1032671168, 33035, 51, 31397381142761241960, 29054157815353374, 166176421788, 645086, 399, 63969560113225176176277, 31426486309136268658, 50672459139597, 25830118, 3113

Offset: 1

Geoffrey Critzer, Oct 15 2012

Row sums are A207828.
Column 1 is A001349.
Column 2 is A216785.
Column 3 is A058915.

			Triangle begins
         1;
         1;
         2,      1;
         6,      2;
        21,      8;
       112,     28,    2;
       853,    145,    7;
     11117,   1022,   34;
    261080,  12320,  181,  1;
  11716571, 274785, 1266, 12;

Needs["Combinatorica`"];max=20;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;CoefficientList[Series[Product[(1+y x^i)^cg[[i]],{i,1,max}],{x,0,max}],{x,y}]//Grid (* after code by Jean-François Alcover in A001349 *)

O.g.f.: Product_{n>=1} (1 + y*x^n)^A001349(n).

A275210 Expansion of (A(x)^2-A(x^2))/2 where A(x) = A001006(x)-1.

Original entry on oeis.org

0, 0, 0, 2, 5, 17, 45, 129, 349, 970, 2658, 7364, 20363, 56634, 157750, 441084, 1236173, 3474672, 9789568, 27648486, 78254719, 221951037, 630717569, 1795576937, 5120472435, 14625574662, 41837913310, 119851980508, 343798008165, 987445317761, 2839518208661

Offset: 0

R. J. Mathar, Jul 19 2016

nonn

Analog of A216785 with Motzkin numbers replacing connected graph counts.

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

Cf. A275209.

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=1..n/2)-
      `if`(n=0 or 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, 1, n/2}] - If[n == 0 || 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) = A275209(2n+1).

A275206 Expansion of (A(x)^2 - A(x^2))/2 where A(x) = A000108(x) - 1.

Original entry on oeis.org

0, 0, 0, 2, 6, 24, 80, 286, 994, 3536, 12576, 45220, 163372, 594320, 2172768, 7983990, 29464010, 109174560, 405990464, 1514797020, 5669004692, 21275014800, 80047213792, 301892460012, 1141068949396, 4321730134624, 16399422014400, 62340424959176, 237373155238104, 905251034394784

Offset: 0

R. J. Mathar, Jul 19 2016

nonn

Number of ways to distribute n pairs of parentheses into 2 distinct patterns where each pattern represents a Catalan ordering (A000108), and each pattern must contain at least one pair of parentheses.
If one of the groups is allowed to have no parentheses, we arrive at A000150 (with a different offset).
Analog of A216785 with Catalan number replacing connected graph counts.
From Petros Hadjicostas, Jul 27 2020: (Start)
It is proved in A050182 that A050182(n) = 1/(2*n + 4)*(binomial(2*n + 4, n) - [(n mod 2) == 0]*binomial(n + 2, n/2)).
Let C(x) = A(x) + 1 = Sum_{n >= 0} c(n)*x^n be the g.f. of the Catalan numbers A000108. Then C(x)^2 = (C(x) - 1)/x. Then (A(x) + 1)^2 = A(x)/x, and thus, A(x)^2 = -2*A(x) - 1 + A(x)/x. Thus, (A(x)^2 - A(x^2))/2 = (-2*A(x) - 1 + A(x)/x - A(x^2))/2.
Substituting A(x) = Sum_{n >= 1} c(n)*x^n in the above expression, we get (after some algebra) that a(n) = (-2*c(n) + c(n+1) - [(n mod 2) == 0]*c(n/2))/2 for n >= 1. It is then easy to prove that a(n) = 2*A050182(n-2) = (1/n)*(binomial(2*n, n-2) - [(n mod 2) == 0]*binomial(n, (n/2) - 1)) for n >= 2, thus proving the conjecture below. (End)

Cf. A000108, A000150, A007223, A050182, A216785.

A[x_] = (1 - Sqrt[1 - 4x])/(2x) - 1;
CoefficientList[(A[x]^2 - A[x^2])/2 + O[x]^30, x] (* Jean-François Alcover, Apr 30 2023 *)

a(2*n+1) = A007223(2*n+1).
Conjecture: a(n) = 2*A050182(n-2) for n >= 2.
From Petros Hadjicostas, Jul 27 2020: (Start)
a(n) = (-2*c(n) + c(n+1) - [(n mod 2) == 0]*c(n/2))/2 for n >= 1, where c = A000108.
a(n) = (1/n)*(binomial(2*n, n-2) - [(n mod 2) == 0]*binomial(n, (n/2) - 1)) for n >= 2. (End)
D-finite with recurrence: 5*(n-3)*(n+2)*(n+1)*a(n) + 32*(n-5)*(n-6)*(2*n-11)*a(n-6) + 80*(n-5)*(n-4)*(n-3)*a(n-5) - 40*(n-4)*(7*n^2-35*n+39)*a(n-4) + (120*n^3 - 600*n^2 + 660*n + 60)*a(n-3) + 2*(n-5)*(23*n^2-44*n+6)*a(n-2) - 5*(n+1)*(7*n^2-25*n+6)*a(n-1) = 0. - Georg Fischer, Feb 12 2025

cp's OEIS Frontend

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

A275166 Number of n-node graphs that have 2 non-isomorphic components.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

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

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A217955 Triangular array read by rows. T(n,k) is the number of unlabeled graphs on n nodes that have exactly k distinct components (n >= 1).

Views

Author

Keywords