A217781 -id:A217781 - cp's OEIS frontend

A027852 Number of connected functions on n points with a loop of length 2.

0, 1, 1, 3, 6, 16, 37, 96, 239, 622, 1607, 4235, 11185, 29862, 80070, 216176, 586218, 1597578, 4370721, 12003882, 33077327, 91433267, 253454781, 704429853, 1962537755, 5479855546, 15332668869, 42983656210, 120716987723, 339596063606, 956840683968

Offset: 1

Christian G. Bower, Dec 14 1997

nonn

Number of unordered pairs of rooted trees with a total of n nodes.
Equivalently, the number of rooted trees on n+1 nodes where the root has degree 2.
Number of trees on n nodes rooted at an edge. - Washington Bomfim, Jul 06 2012
Guy (1988) calls these tadpole graphs. - N. J. A. Sloane, Nov 04 2014
Number of unicyclic graphs of n nodes with a cycle length of two (in other words, a double edge). - Washington Bomfim, Dec 02 2020

Alois P. Heinz, Table of n, a(n) for n = 1..2136
Washington Bomfim, Illustration of initial terms
R. K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy) Includes illustrations for n <= 6.
R. J. Mathar, Topologically distinct sets of non-intersecting circles in the plane, arXiv:1603.00077 [math.CO] (2016), Eq. (75).
Index entries for sequences related to rooted trees

Column 2 of A033185 (forests of rooted trees), A217781 (unicyclic graphs), A339303 (unoriented linear forests) and A339428 (connected functions).

Cf. A000081, A000106, A000226, A000631, A001372, A002861.

with(numtheory): b:= proc(n) option remember; local d, j; `if`(n<=1, n, (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/ (n-1)) end: a:= n-> (add(b(i) *b(n-i), i=0..n) +`if`(irem(n, 2)=0, b(n/2), 0))/2: seq(a(n), n=1..50);  # Alois P. Heinz, Aug 22 2008, revised Oct 07 2011
# second, re-usable version
A027852 := proc(N::integer)
    local dh, Nprime;
    dh := 0 ;
    for Nprime from 0 to N do
        dh := dh+A000081(Nprime)*A000081(N-Nprime) ;
    end do:
    if type(N,'even') then
        dh := dh+A000081(N/2) ;
    end if;
    dh/2 ;
end proc: # R. J. Mathar, Mar 06 2017

Needs["Combinatorica`"];nn = 30; s[n_, k_] := s[n, k] = a[n + 1 - k] + If[n < 2 k, 0, s[n - k, k]]; a[1] = 1; a[n_] := a[n] = Sum[a[i] s[n - 1, i] i, {i, 1, n - 1}]/(n - 1); rt = Table[a[i], {i, 1, nn}]; Take[CoefficientList[CycleIndex[DihedralGroup[2], s] /. Table[s[j] -> Table[Sum[rt[[i]] x^(k*i), {i, 1, nn}], {k, 1, nn}][[j]], {j, 1, nn}], x], {2, nn}]  (* Geoffrey Critzer, Oct 12 2012, after code given by Robert A. Russell in A000081 *)
b[n_] := b[n] = If[n <= 1, n, (Sum[Sum[d b[d], {d, Divisors[j]}] b[n-j], {j, 1, n-1}])/(n-1)];
a[n_] := (Sum[b[i] b[n-i], {i, 0, n}] + If[Mod[n, 2] == 0, b[n/2], 0])/2;
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Oct 30 2018, after Alois P. Heinz *)

seq(max_n)= { my(V = f = vector(max_n), i=1,s); f[1]=1;
for(j=1, max_n - 1, f[j+1] = 1/j * sum(k=1, j, sumdiv(k,d, d * f[d]) * f[j-k+1]));
for(n = 1, max_n, s = sum(k = 1, (n-1)/2, ( f[k] * f[n-k] ));
if(n % 2 == 1, V[i] = s, V[i] = s + (f[n/2]^2 + f[n/2])/2); i++); V };
\\ Washington Bomfim, Jul 06 2012 and Dec 01 2020

G.f.: A(x) = (B(x)^2 + B(x^2))/2 where B(x) is g.f. of A000081.
a(n) = Sum_{k=1..(n-1)/2}( f(k)*f(n-k) ) + [n mod 2 = 0] * ( f(n/2)^2+f(n/2) ) /2, where f(n) = A000081(n). - Washington Bomfim, Jul 06 2012 and Dec 01 2020
a(n) ~ c * d^n / n^(3/2), where d = A051491 = 2.9557652856519949747148..., c = A187770 = 0.43992401257102530404090339... . - Vaclav Kotesovec, Sep 12 2014
2*a(n) = A000106(n) + A000081(n/2), where A(.)=0 if the argument is non-integer. - R. J. Mathar, Jun 04 2020

Edited by Christian G. Bower, Feb 12 2002

A339428 Triangle read by rows: T(n,k) is the number of connected functions on n points with a loop of length k.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 9, 6, 3, 1, 1, 20, 16, 9, 4, 1, 1, 48, 37, 23, 11, 4, 1, 1, 115, 96, 62, 35, 14, 5, 1, 1, 286, 239, 169, 97, 46, 18, 5, 1, 1, 719, 622, 451, 282, 145, 63, 21, 6, 1, 1, 1842, 1607, 1217, 792, 440, 206, 80, 25, 6, 1, 1

Offset: 1

Andrew Howroyd, Dec 03 2020

			Triangle begins:
    1;
    1,   1;
    2,   1,   1;
    4,   3,   1,   1;
    9,   6,   3,   1,   1;
   20,  16,   9,   4,   1,  1;
   48,  37,  23,  11,   4,  1,  1;
  115,  96,  62,  35,  14,  5,  1, 1;
  286, 239, 169,  97,  46, 18,  5, 1, 1;
  719, 622, 451, 282, 145, 63, 21, 6, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Columns 1..12 are A000081, A027852, A029852, A029853, A029868, A029869, A029870, A029871, A032205, A032206, A032207, A032208.
Row sums are A002861.
Cf. A033185, A217781, A339067.

\\ TreeGf is A000081 as g.f.
TreeGf(N) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
ColSeq(n,k)={my(r=TreeGf(max(0,n+1-k))); Vec(sumdiv(k, d, eulerphi(d)*subst(r + O(x*x^(n\d)), x, x^d)^(k/d))/k, -n)}
M(n, m=n)=Mat(vector(m, k, ColSeq(n,k)~))
{ my(T=M(12)); for(n=1, #T~, print(T[n,1..n])) }

G.f. of k-th column: (1/k)*Sum_{d|k} phi(d) * r(x^d)^(k/d) where r(x) is the g.f. of A000081.

A339067 Triangle read by rows: T(n,k) is the number of linear forests with n nodes and k rooted trees.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 9, 12, 9, 4, 1, 20, 30, 25, 14, 5, 1, 48, 74, 69, 44, 20, 6, 1, 115, 188, 186, 133, 70, 27, 7, 1, 286, 478, 503, 388, 230, 104, 35, 8, 1, 719, 1235, 1353, 1116, 721, 369, 147, 44, 9, 1, 1842, 3214, 3651, 3168, 2200, 1236, 560, 200, 54, 10, 1

Offset: 1

Andrew Howroyd, Dec 03 2020

T(n,k) is the number of trees with n nodes rooted at two noninterchangeable nodes at a distance k-1 from each other.

Also the convolution triangle of A000081. - Peter Luschny, Oct 07 2022

			Triangle begins:
    1;
    1,    1;
    2,    2,    1;
    4,    5,    3,    1;
    9,   12,    9,    4,   1;
   20,   30,   25,   14,   5,   1;
   48,   74,   69,   44,  20,   6,   1;
  115,  188,  186,  133,  70,  27,   7,  1;
  286,  478,  503,  388, 230, 104,  35,  8, 1;
  719, 1235, 1353, 1116, 721, 369, 147, 44, 9, 1;
  ...

Alois P. Heinz, Rows n = 1..200, flattened

Columns 1..6 are A000081, A000106, A000242, A000300, A000343, A000395.
Row sums are A000107.
T(2n-1,n) gives A339440.
Cf. A033185, A038002, A217781, A339428.

b:= proc(n) option remember; `if`(n<2, n, (add(add(d*b(d),
      d=numtheory[divisors](j))*b(n-j), j=1..n-1))/(n-1))
    end:
T:= proc(n, k) option remember; `if`(k=1, b(n), (t->
      add(T(j, t)*T(n-j, k-t), j=1..n-1))(iquo(k, 2)))
    end:
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Dec 04 2020
# Using function PMatrix from A357368. Adds row and column for n, k = 0.
PMatrix(10, A000081); # Peter Luschny, Oct 07 2022

b[n_] := b[n] = If[n < 2, n, (Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n - j], {j, 1, n - 1}])/(n - 1)];
T[n_, k_] := T[n, k] = If[k == 1, b[n], With[{t = Quotient[k, 2]}, Sum[T[j, t]*T[n - j, k - t], {j, 1, n - 1}]]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jan 03 2021, after Alois P. Heinz *)

\\ TreeGf is A000081.
TreeGf(N) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
ColSeq(n,k)={my(t=TreeGf(max(0,n+1-k))); Vec(t^k, -n)}
M(n, m=n)=Mat(vector(m, k, ColSeq(n,k)~))
{ my(T=M(12)); for(n=1, #T~, print(T[n,1..n])) }

G.f. of k-th column: t(x)^k where t(x) is the g.f. of A000081.

Sum_{k=1..n} k * T(n,k) = A038002(n). - Alois P. Heinz, Dec 04 2020

A000226 Number of n-node unlabeled connected graphs with one cycle of length 3.

Original entry on oeis.org

1, 1, 3, 7, 18, 44, 117, 299, 793, 2095, 5607, 15047, 40708, 110499, 301541, 825784, 2270211, 6260800, 17319689, 48042494, 133606943, 372430476, 1040426154, 2912415527, 8167992598, 22947778342, 64577555147, 182009003773, 513729375064, 1452007713130

Offset: 3

N. J. A. Sloane

Number of rooted trees on n+1 nodes where root has degree 3. - Christian G. Bower
Third column of A033185. - Michael Somos, Aug 20 2018
From Washington Bomfim, Dec 22 2020: (Start)
Number of forests of 3 rooted trees with a total of n nodes.
Number of unicyclic graphs with a cycle of length 3 and a total of n nodes.
(End)

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
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).

Alois P. Heinz, Table of n, a(n) for n = 3..1000 (first 198 terms from Vincenzo Librandi)
Sergei Abramovich, Partitions of integers, Ferrers-Young diagrams, and representational efficacy of spreadsheet modeling, Spreadsheets in Education (eJSiE): Vol. 5: Iss. 2, Article 1, p. 5
Washington Bomfim, Illustration of initial terms
F. Ruskey, Combinatorial Generation, Eq.(4.27), 2003
Eric Weisstein's World of Mathematics, Rooted Tree
Index entries for sequences related to rooted trees

Column 3 of A033185 and A217781.
Cf. A000081, A001429.
For n >= 3 a(n) = A217781(n, 3) = A058879(n, n-2) = A033185(n, 3).

b:= proc(n) option remember; if n<=1 then n else add(k*b(k)* s(n-1, k), k=1..n-1)/(n-1) fi end: s:= proc(n,k) option remember; add(b(n+1-j*k), j=1..iquo(n,k)) end: B:= proc(n) option remember; unapply(add(b(k)*x^k, k=1..n),x) end: a:= n-> coeff(series((B(n-2)(x)^3+ 3*B(n-2)(x)* B(n-2)(x^2)+ 2*B(n-2)(x^3))/6, x=0, n+1), x,n): seq(a(n), n=3..40); # Alois P. Heinz, Aug 21 2008

terms = 30; r[] = 0; Do[r[x] = x *Exp[Sum[r[x^k]/k, {k, 1, j}]] + O[x]^j // Normal, {j, 1, terms+3}]; A[x_] = (r[x]^3 + 3*r[x]*r[x^2] + 2*r[x^3])/6 + O[x]^(terms+3); Drop[CoefficientList[A[x], x], 3] (* Jean-François Alcover, Nov 23 2011, updated Jan 11 2018 *)

seq(max_n) = {my(a = f = vector(max_n), s, D); f[1]=1;
for(j=1, max_n - 1, f[j+1] = 1/j * sum(k=1, j, sumdiv(k,d, d * f[d]) * f[j-k+1]));
for(n=3,max_n,s=0;forpart(P=n,D=Set(P);if(#D==3,s+=f[P[1]]*f[P[2]]*f[P[3]];next());
if(#D==1, s+= binomial(f[P[1]]+2, 3); next());
if(P[1] == P[2], s += binomial(f[P[1]]+1, 2) * f[P[3]],
s += binomial(f[P[2]]+1, 2) * f[P[1]]),[1,n],[3,3]); a[n] = s ); a[3..max_n] }; \\ Washington Bomfim, Dec 22 2020

G.f.: (r(x)^3+3*r(x)*r(x^2)+2*r(x^3))/6 where r(x) is g.f. for rooted trees (A000081).
a(n) = Sum_{j1+2j2+···= n} (Product_{i=1..n} binomial(A000081(i) + j_i -1, j_i)) [(4.27) of [F. Ruskey] with n replaced by n+1]. - Washington Bomfim, Dec 22 2020
a(n) ~ (A187770 + A339986) * A051491^n / (2 * n^(3/2)). - Vaclav Kotesovec, Dec 25 2020

More terms from Vladeta Jovovic, Apr 19 2000

A000368 Number of connected graphs with one cycle of length 4.

Original entry on oeis.org

1, 1, 4, 9, 28, 71, 202, 542, 1507, 4114, 11381, 31349, 86845, 240567, 668553, 1860361, 5188767, 14495502, 40572216, 113743293, 319405695, 898288484, 2530058013, 7135848125, 20152898513, 56986883801

Offset: 4

N. J. A. Sloane

nonn

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973, page 69.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
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).

Vaclav Kotesovec, Table of n, a(n) for n = 4..2000 (terms 4..43 from Sean A. Irvine, 44..200 from Washington Bomfim)
Washington Bomfim, Illustration of initial terms

Column k=4 of A217781.
Cf. A000081, A000226, A001429, A005703.
Second diagonal of A058879.

Needs["Combinatorica`"]; nn = 30; s[n_, k_] := s[n, k] = a[n + 1 - k] + If[n < 2 k, 0, s[n - k, k]]; a[1] = 1; a[n_] := a[n] = Sum[a[i] s[n - 1, i] i, {i, 1, n - 1}]/(n - 1); rt = Table[a[i], {i, 1, nn}]; Take[CoefficientList[CycleIndex[DihedralGroup[4], s] /. Table[s[j] -> Table[Sum[rt[[i]] x^(k*i), {i, 1, nn}], {k, 1, nn}][[j]], {j, 1, nn}], x], {5, nn}]  (* Geoffrey Critzer, Oct 12 2012, after code given by Robert A. Russell in A000081 *)
A000081 = Rest[Cases[ Import["https://oeis.org/A000081/b000081.txt", "Table"], {, }][[All, 2]]]; max = 30; g81 = Sum[A000081[[k]]*x^k, {k, 1, max}]; g81x2 = Sum[A000081[[k]]*x^(2 k), {k, 1, max}]; g81x4 = Sum[A000081[[k]]*x^(4 k), {k, 1, max}]; Drop[CoefficientList[ Series[(2*g81x4 + 3*g81x2^2 + 2*g81^2*g81x2 + g81^4)/8, {x, 0, max}], x], 4] (* Vaclav Kotesovec, Dec 25 2020 *)

g(Q)={my(V=Vec(Q),D=Set(V),d=#D); if(d==4,return(3*f[D[1]]*f[D[2]]*f[D[3]]*f[D[4]]));
if(d==1, return((f[D[1]]^4+2*f[D[1]]^3+3*f[D[1]]^2+2*f[D[1]])/8));
my(k=1, m = #select(x->x == D[k],V), t); while(m==1, k++; m = #select(x->x == D[k], V)); t = D[1]; D[1] = D[k]; D[k] = t;
if(d == 3, return( f[D[1]] * f[D[2]] * f[D[3]] * (3 * f[D[1]] + 1)/2 ) );
if(m==3, return(f[D[1]]^2 * f[D[2]] * (f[D[1]] + 1)/2));
((3*f[D[2]]^2 + f[D[2]])*f[D[1]]^2 + (f[D[2]]^2 + 3*f[D[2]])*f[D[1]])/4 };
seq(max_n) = { my(s, a = vector(max_n), U); f = vector(max_n); f[1] = 1;
for(j=1, max_n - 1, if(j%100==0,print(j)); f[j+1] = 1/j * sum(k=1, j, sumdiv(k,d, d * f[d]) * f[j-k+1]));
for(n=4, max_n, s=0; forpart(Q = n, if( (Q[4] > Q[3]) && (Q[3]-1 > Q[2]),
      U = U / (f[Q[4] + 1] * f[Q[3] - 1]) * f[Q[4]] * f[Q[3]],  U = g(Q)); s += U,
[1,n],[4,4]); a[n] = s; if(n % 100 == 0, print(n": " s))); a[4..max_n] };
\\ Washington Bomfim, Jul 19 2012 and Dec 22 2020

From Washington Bomfim, Jul 19 2012 and Dec 22 2020: (Start)
a(n) = Sum_{P}( g(Q) ), where P is the set of the partitions Q of n with 4 parts, Q with distinct parts D[1]..D[d], D[1] the part of maximum multiplicity m in Q, f(n) = A000081(n), and g(Q) given by,
       | 3 * f(D[1]) * f(D[2]) * f(D[3]) * f(D[4]),              if d = 4,
       | (f(D[1])^4 + 2*f(D[1])^3 + 3*f(D[1])^2 + 2*f(D[1]))/8,  if d = 1,
g(Q) = | f(D[1]) * f(D[2]) * f(D[3]) * (3 * f(D[1]) + 1)/2,      if d = 3,
       | ((3*f(D[2])^2+f(D[2]))*f(D[1])^2+(f(D[2])^2+3*f(D[2]))*f(D[1]))/4,
       |                                                   if d=2, and m=2,
       | f(D[1])^2 * f(D[2]) * (f(D[1]) + 1)/2,            if d=2, and m=3.
(End)
G.f.: (2*t(x^4) + 3*t(x^2)^2 + 2*t(x)^2*t(x^2) + t(x)^4)/8 where t(x) is the g.f. of A000081. - Andrew Howroyd, Dec 03 2020
a(n) ~ (A187770 + A339986) * A051491^n / (2 * n^(3/2)). - Vaclav Kotesovec, Dec 25 2020

More terms from Vladeta Jovovic, Apr 20 2000
Definition improved by Franklin T. Adams-Watters, May 16 2006
More terms from Sean A. Irvine, Nov 14 2010

A068051 Number of n-node connected graphs with one cycle, possibly of length 1 or 2.

Original entry on oeis.org

1, 2, 4, 9, 20, 49, 118, 300, 765, 1998, 5255, 14027, 37670, 102095, 278262, 763022, 2101905, 5816142, 16153148, 45017423, 125836711, 352723949, 991143727, 2791422887, 7877935985, 22275473767, 63096075118, 179012076933

Offset: 1

Christian G. Bower, Feb 12 2002

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
C. G. Bower, Transforms (2)

Cf. A217781.

nn=20;t[x_]:=Sum[a[n]x^n,{n,0,nn}];sol=SolveAlways[0==Series[t[x]-x Product[1/(1-x^i)^a[i],{i,1,nn}],{x,0,nn}],x];b=Table[a[n],{n,1,nn}]/.sol//Flatten;Map[Total,Drop[Transpose[Table[Take[CoefficientList[CycleIndex[DihedralGroup[n],s]/.Table[s[j]->Table[Sum[b[[i]]x^(i*k),{i,1,nn}],{k,1,nn}][[j]],{j,1,n}],x],nn],{n,1,nn}]],1]]  (* Geoffrey Critzer, Mar 24 2013 *)

\\ TreeGf gives gf of A000081
TreeGf(N)={my(A=vector(N, j, 1)); for(n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
seq(n)={my(t=TreeGf(n)); my(g(e)=subst(t + O(x*x^(n\e)), x, x^e) + O(x*x^n)); Vec((sum(d=1, n, eulerphi(d)/d*log(1/(1-g(d)))) + ((1+g(1))^2/(1-g(2))-1)/2)/2)} \\ Andrew Howroyd, Jun 20 2018

"DIK" transform of A000081.

a(n) = A000081(n) + A027852(n) + A000226(n) + A000368(n) + ... [Geoffrey Critzer, Mar 24 2013]

A000631 Number of ethylene derivatives with n carbon atoms.

Original entry on oeis.org

1, 1, 3, 5, 13, 27, 66, 153, 377, 914, 2281, 5690, 14397, 36564, 93650, 240916, 623338, 1619346, 4224993, 11062046, 29062341, 76581151, 202365823, 536113477, 1423665699, 3788843391, 10103901486, 26995498151, 72253682560, 193706542776

Offset: 2

N. J. A. Sloane

nonn

Number of structural isomers of alkenes C_n H_{2n} with n carbon atoms.
Number of unicyclic graphs of n nodes where a double-edge replaces the cycle, [A217781], end-points of the double-edge of out-degrees <= 2, other nodes having out-degrees <= 3.
Number of rooted trees on n+1 nodes where the root has degree 2, the 2 children of the root have out-degrees <= 2, and the other nodes have out-degrees <= 3.
See illustration of initial terms. - Washington Bomfim, Nov 30 2020

J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
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).

Andrew Howroyd, Table of n, a(n) for n = 2..500
Washington Bomfim, Illustration of initial terms
J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
H. R. Henze and C. M. Blair, The number of structurally isomeric Hydrocarbons of the Ethylene Series, J. Amer. Chem. Soc., 55 (2) (1933), 680-686.
H. R. Henze and C. M. Blair, The number of structurally isomeric Hydrocarbons of the Ethylene Series, J. Amer. Chem. Soc., 55 (2) (1933), 680-685. (Annotated scanned copy)
H. R. Henze and C. M. Blair, The number of structural isomers of the more important types of aliphatic compounds, J. Amer. Chem. Soc., 56 (1) (1934), 157-157.
C.-W. Lam, A Mathematical Relationship between the Number of Isomers of Alkenes and Alkynes: A Result Established from the Enumeration of Isomers of Alkenes from Alky Biradicals, J. Math. Chem., 23, 421 (1998).
R. J. Mathar, Illustration for graphs up to 6 carbons
R. C. Read, Some recent results in chemical enumeration, Lect. Notes Math. 303 (1972), 243-259.
R. C. Read, Some recent results in chemical enumeration, Preprint, circa 1972. (Annotated scanned copy)
R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976. [Annotated scanned copy] See p. 28.
N. Trinajstich, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski, Computer Generation of Isomeric Structures, Pure & Appl. Chem., Vol. 55, No. 2, pp. 379-390, 1983.
Index entries for sequences related to rooted trees

Cf. A000642, A000598, A027852 (out-degrees of nodes not limited).

\\ Here G(n) is A000598 as g.f., h is A000642.
seq(n)={my(g=G(n), h=(subst(g, x, x^2) + g^2)/2); Vec(subst(h, x, x^2) + h^2)/2} \\ Andrew Howroyd, Dec 01 2020

a(n) = b(1)b(n-1) + b(2)b(n-2) + b(3)b(n-3) + ... + b(n/2)(b(n/2) + 1)/2 when n is even or b(1)b(n-1) + b(2)b(n-2) + b(3)b(n-3) + ... + b((n-1)/2)b((n + 1)/2) when n is odd, where b(n) = A000642(n). - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 24 2008
a(n) = Sum_{k=1..(n-1)/2}( f(k) * f(n-k) ) + [n mod 2 = 0] * ( f(n/2)^2 + f(n/2) ) / 2 where f(n) = A000642(n+1). - Washington Bomfim, Nov 29 2020
G.f.: (g(x^2) + g(x)^2)/2 where x*g(x) is the g.f. of A000642. - Andrew Howroyd, Dec 01 2020

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 24 2008

A058879 Triangle read by rows: T(n,k) = number of connected graphs with one cycle of length m = n - k + 1 and n nodes (n >= 3 and 1 <= k <= n - 2).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 4, 7, 1, 1, 4, 9, 18, 1, 1, 5, 10, 28, 44, 1, 1, 5, 13, 32, 71, 117, 1, 1, 6, 14, 45, 89, 202, 299, 1, 1, 6, 17, 52, 130, 264, 542, 793, 1, 1, 7, 19, 69, 163, 413, 751, 1507, 2095, 1, 1, 7, 22, 79, 224, 544, 1221, 2179, 4114, 5607

Offset: 3

N. J. A. Sloane, Jan 07 2001

Diagonals give A000226, A000368. Row sums give A001429.
From Washington Bomfim, Jul 06 2008: (Start)
T(n,3) = 2 + floor(m/2). When k = 3, n = m + 2, so we have unicyclic graphs of order m+2 with a cycle of length m. Only two nodes of those graphs belong to the rooted trees attached to the cycle, so the orders of those trees can be only 1, 2, or 3.
We can have only one tree of order 3 in those graphs. So the two different rooted trees of order 3 correspond to two unicycles.
We can have two trees of order 2 in those graphs. Those trees can be rooted at two points r_1, r_2 of the cycle in h = floor(m/2) ways. They can be neighbors, i.e., we have an edge of the cycle (r_1, r_2). They can be 2, 3, ..., h edges apart, but they cannot be h+1 edges away from each other. This is true because we obtain an isomorphic graph if r_1 and r_2 are h+1 (or more) edges apart, since there are also n - (h+1) edges between r_1 and r_2 and n-h-1 <= h. Note that there is only one rooted tree of order two.
The five unicyclic graphs of order 9 with a cycle of length 7 are depicted in the picture corresponding to the link.
T(n,4) = 4 + 2floor(m/2) + nearest integer to m^2/12.
We have unicyclic graphs of order m+3 with a cycle of length m. Only three nodes of those graphs belong to the rooted trees attached to the cycle, so the orders of those trees can be only 1, 2, 3, or 4. The set of unicycles can be divided in graphs with trees of orders
4,1,1,...,1
3,2,1,...,1
2,2,2,1,...,1.
Since there are 4 rooted trees of order 4, the orders 4,1,1,...,1 correspond to 4 unicycles.
The orders 3,2,1,...,1 correspond to 2floor(m/2) unicycles. For each one of the two rooted trees of order 3, we see above that there are floor(m/2) possibilities to choose a root for the tree of order 2.
The orders 2,2,2,1,...,1 correspond to i unicycles, i = nearest integer to m^2/12. This follows from the number of necklaces with n+3 beads 3 of which are red, that is equal to the nearest integer to (n+3)^2/12. See A001399. In our case we have necklaces with m beads. The 3 red beads are the roots of the trees of order 2. (End)

			Triangle begins:
  1;
  1,  1;
  1,  1,  3;
  1,  1,  4,  7;
  1,  1,  4,  9, 18;

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 69, (3.4.1).
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150, Table 9.

Washington Bomfim, The five unicyclic graphs of order 9 with a cycle of length 7: T(9,3)=5.

Cf. A000081, A217781.

Needs["NumericalDifferentialEquationAnalysis`"];
t[n_, k_] := Block[{x}, Coefficient[CycleIndexPolynomial[DihedralGroup[n + 1 - k], Table[ButcherTreeCount[n].x^(p Range[n]), {p, n + 1 - k}]], x, n]];
Table[t[n, k], {n, 13}, {k, 1, n - 2}] // Flatten
(* requires Mathematica 9+; Andrey Zabolotskiy, May 12 2017 *)

T(n, k) = [x^n] Z(D_{n+1-k}; t(x)) where t(x) is the g.f. of A000081 and Z(D_m) is the cycle index of the dihedral group of order m. - Sean A. Irvine, Sep 03 2022

More terms from Washington Bomfim, May 12 2008
More terms from Washington Bomfim, Jul 06 2008
Rows n = 11 to 13 added, name and offset corrected by Andrey Zabolotskiy, May 12 2017

A339984 G.f.: g(x) * g(x^2), where g(x) is the g.f. of A000081.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 5, 13, 26, 65, 147, 369, 899, 2298, 5851, 15261, 39945, 105948, 282504, 759480, 2052027, 5576017, 15216998, 41705762, 114715503, 316611401, 876466003, 2433091773, 6771462322, 18889829555, 52809592990, 147935027381, 415182991401, 1167251435240

Offset: 0

Vaclav Kotesovec, Dec 25 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A000081, A217781, A309619, A339985.

max = 30; A000081 = Rest[Cases[ Import["https://oeis.org/A000081/b000081.txt", "Table"], {, }][[All, 2]]]; g81 = Sum[A000081[[k]]*x^k, {k, 1, max}]; g81x2 = Sum[A000081[[k]]*x^(2*k), {k, 1, max}]; CoefficientList[Series[g81 * g81x2, {x, 0, max}], x]

a(n) ~ A339986 * A051491^n / n^(3/2).

A339985 G.f.: g(x)^2 * g(x^2), where g(x) is the g.f. of A000081.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 6, 14, 37, 90, 232, 584, 1512, 3906, 10246, 26984, 71766, 191852, 516400, 1396760, 3797435, 10367628, 28420466, 78183462, 215791426, 597368222, 1658233794, 4614679792, 12872125836, 35982713314, 100787606966, 282832173830, 795070060983

Offset: 0

Vaclav Kotesovec, Dec 25 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A000081, A217781, A339984.

max = 30; A000081 = Rest[Cases[ Import["https://oeis.org/A000081/b000081.txt", "Table"], {, }][[All, 2]]]; g81 = Sum[A000081[[k]]*x^k, {k, 1, max}]; g81x2 = Sum[A000081[[k]]*x^(2*k), {k, 1, max}]; CoefficientList[Series[g81^2 * g81x2, {x, 0, max}], x]

a(n) ~ 2 * A339986 * A051491^n / n^(3/2).

cp's OEIS Frontend

A027852 Number of connected functions on n points with a loop of length 2.

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A339428 Triangle read by rows: T(n,k) is the number of connected functions on n points with a loop of length k.

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A339067 Triangle read by rows: T(n,k) is the number of linear forests with n nodes and k rooted trees.

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A000226 Number of n-node unlabeled connected graphs with one cycle of length 3.

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A000368 Number of connected graphs with one cycle of length 4.

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A068051 Number of n-node connected graphs with one cycle, possibly of length 1 or 2.

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A000631 Number of ethylene derivatives with n carbon atoms.

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