A000598 -id:A000598 - cp's OEIS frontend

A292555 Number of rooted unlabeled trees on n nodes where each node has at most 10 children.

1, 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4765, 12483, 32964, 87785, 235305, 634628, 1720524, 4686842, 12820920, 35206475, 97010705, 268154003, 743351390, 2066090876, 5756490561, 16074597300, 44980514021, 126109353817, 354202275766, 996517941454

Offset: 0

Marko Riedel, Sep 18 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Marko Riedel, Trees with bounded degree.

Cf. A000081, A001190, A000598, A036718, A036721, A036722, A182378, A244372, A292553, A292554, A292556.

Column k=10 of A299038.

b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
       b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
    end:
a:= n-> `if`(n=0, 1, b(n-1$2, 10$2)):
seq(a(n), n=0..35);  # Alois P. Heinz, Sep 20 2017

b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[ Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, t - j, k], {j, 0, Min[t, n/i]}]]];
a[n_] := If[n == 0, 1, b[n - 1, n - 1, 10, 10]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

Functional equation of G.f. is T(z) = z + z*Sum_{q=1..10} Z(S_q)(T(z)) with Z(S_q) the cycle index of the symmetric group. Alternate FEQ is
T(z) = 1 + z*Z(S_10)(T(z)).
a(n) = Sum_{j=1..10} A244372(n,j) for n>0, a(0) = 1. - Alois P. Heinz, Sep 20 2017
a(n) / a(n+1) ~ 0.338329194566131211670667671160855741193081902868090986608524... - Robert A. Russell, Feb 11 2023

A036671 Number of isomers C_n H_{2n} without double bonds.

Original entry on oeis.org

0, 0, 1, 2, 5, 12, 29, 73, 185, 475, 1231, 3232, 8506, 22565, 60077, 160629, 430724, 1158502, 3122949, 8437289, 22836877, 61918923, 168139339, 457225555, 1244935251, 3393754661, 9261681937, 25301337669, 69184724389, 189349490641

Offset: 1

N. J. A. Sloane

Equivalently, the number of simple unicyclic graphs on n unlabeled vertices with all degrees at most 4. See table 1 in Michael A. Kappler reference. - Jonathan Vos Post, Dec 07 2005, Andrew Howroyd, May 22 2018

Camden A. Parks and James B. Hendrickson, Enumeration of monocyclic and bicyclic carbon skeletons, J. Chem. Inf. Comput. Sci., vol. 31, 334-339 (1991). See page 335 Table 1.
J. B. Hendrikson and C. A. Parks, "Generation and Enumeration of Carbon skeletons", J. Chem. Inf. Comput. Sci, vol. 31 (1991) pp. 101-107. See Table 2, column 3 on page 103.

Andrew Howroyd, Table of n, a(n) for n = 1..500
Michael A. Kappler, GENSMI: Exhaustive Enumeration of Simple Graphs.
G. Polya, Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen, Acta Math. 68 (1937), 145-254.

Cf. A000598, A000642, A001429.

\\ here G is A000598 as series
G(n)={my(g=O(x)); for(n=1, n, g = 1 + x*(g^3/6 + subst(g,x,x^2)*g/2 + subst(g,x,x^3)/3) + O(x^n)); g}
seq(n)={my(t=G(n-2)); t=x*(t^2+subst(t,x,x^2))/2; my(g(e)=subst(t + O(x*x^(n\e)), x, x^e) + O(x*x^n)); Vec(sum(k=3, n, sumdiv(k, d, eulerphi(d)*g(d)^(k/d))/k + if(k%2, g(1)*g(2)^(k\2), (g(1)^2+g(2))*g(2)^(k/2-1)/2))/2, -n)} \\ Andrew Howroyd, May 22 2018

Polya reference gives an explicit g.f.; so does Parks et al.

More terms from Vladeta Jovovic, Aug 19 2001

A000200 Number of bicentered hydrocarbons with n atoms.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 3, 9, 15, 38, 73, 174, 380, 915, 2124, 5134, 12281, 30010, 73401, 181835, 452165, 1133252, 2851710, 7215262, 18326528, 46750268, 119687146, 307528889, 792716193, 2049703887, 5314775856, 13817638615, 36012395538

Offset: 0

N. J. A. Sloane, E. M. Rains (rains(AT)caltech.edu)

Busacker and Saaty, Finite Graphs and Networks, 1965, p. 201 (they reproduce Cayley's mistakes).
A. Cayley, "On the mathematical theory of isomers", Phil. Mag. vol. 67 (1874), 444-447.
A. Cayley, "Über die analytischen Figuren, welche in der Mathematik Baeume genannt werden...", Chem. Ber. 8 (1875), 1056-1059.
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, Table of n, a(n) for n = 0..60
Jean-François Alcover, Mathematica program translated from N. J. A. Sloane's Maple program for A000022, A000200, A000598, A000602, A000678
Henry Bottomley, Illustration of initial terms of A000022, A000200, A000602
A. Cayley, Über die analytischen Figuren, welche in der Mathematik Bäume genannt werden und ihre Anwendung auf die Theorie chemischer Verbindungen, Chem. Ber. 8 (1875), 1056-1059. (Annotated scanned copy)
E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees), J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
N. J. A. Sloane, Maple program and first 60 terms for A000022, A000200, A000598, A000602, A000678
Index entries for sequences related to trees

A000200 = A000602 - A000022 for n>0.

Cf. A010373.

N := 45: for i from 1 to N do tt := t[ i ]-t[ i-1 ]; b[ i ] := series((tt^2+subs(z=z^2,tt))/2+O(z^(N+1)),z,200): od: i := 'i': bicent := series(sum(b[ i ],i=1..N),z,200); G000200 := bicent; A000200 := n->coeff(G000200,z,n);
# Maple code continues from A000022: bicentered == unordered pair of ternary trees of the same height:

n = 40; (* algorithm from Rains and Sloane *)
S3[f_,h_,x_] := f[h,x]^3/6 + f[h,x] f[h,x^2]/2 + f[h,x^3]/3;
T[-1,z_] := 1;  T[h_,z_] := T[h,z] = Table[z^k, {k,0,n}].Take[CoefficientList[z^(n+1) + 1 + S3[T,h-1,z]z, z], n+1];
Sum[Take[CoefficientList[z^(n+1) + (T[h,z] - T[h-1,z])^2/2 + (T[h,z^2] - T[h-1,z^2])/2, z],n+1], {h,0,n/2}] (* Robert A. Russell, Sep 15 2018 *)

A295461 Number of unlabeled rooted trees with 2n + 1 nodes in which all outdegrees are even.

Original entry on oeis.org

1, 1, 2, 5, 12, 33, 91, 264, 780, 2365, 7274, 22727, 71784, 229094, 737215, 2390072, 7798020, 25587218, 84377881, 279499063, 929556155, 3102767833, 10390936382, 34903331506, 117564309276, 396994228503, 1343716120550, 4557952756658, 15491856887741

Offset: 0

Gus Wiseman, Jan 13 2018

nonn

			The a(3) = 5 trees: (o(o(oo))), (o(oooo)), ((oo)(oo)), (ooo(oo)), (oooooo).

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

Cf. A000081, A000598, A001190, A003238, A004111, A027193, A027187, A032305, A067659, A290689, A291443, A297791, A298118, A298120, A298126.

erut[n_]:=erut[n]=If[n===1,{{}},Join@@Function[c,Union[Sort/@Tuples[erut/@c]]]/@Select[IntegerPartitions[n-1],EvenQ[Length[#]]&]];
Table[Length[erut[n]],{n,1,30,2}]

A305059 Triangle read by rows: T(n,k) is the number of connected unicyclic graphs on n unlabeled nodes with cycle length k and all nodes having degree at most 4.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 6, 4, 1, 1, 15, 8, 4, 1, 1, 33, 24, 9, 5, 1, 1, 83, 55, 28, 12, 5, 1, 1, 196, 147, 71, 40, 13, 6, 1, 1, 491, 365, 198, 106, 47, 16, 6, 1, 1, 1214, 954, 521, 317, 136, 63, 18, 7, 1, 1, 3068, 2431, 1418, 868, 428, 190, 73, 21, 7, 1, 1

Offset: 3

Andrew Howroyd, May 24 2018

Equivalently, the number of monocyclic skeletons with n carbon atoms and a ring size of k.

			Triangle begins:
     1;
     1,   1;
     3,   1,   1;
     6,   4,   1,   1;
    15,   8,   4,   1,   1;
    33,  24,   9,   5,   1,  1;
    83,  55,  28,  12,   5,  1,  1;
   196, 147,  71,  40,  13,  6,  1, 1;
   491, 365, 198, 106,  47, 16,  6, 1, 1;
  1214, 954, 521, 317, 136, 63, 18, 7, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 3..1178 (rows n = 3..50)
Camden A. Parks and James B. Hendrickson, Enumeration of monocyclic and bicyclic carbon skeletons, J. Chem. Inf. Comput. Sci., vol. 31, 334-339 (1991).
Eric Weisstein's World of Mathematics, Unicyclic Graph

Columns 3..10 are A063832, A116719, A120333, A120779, A120790, A120795, A121156, A121157.
Row sums are A036671.
Cf. A000598.

G[n_] := Module[{g}, Do[g[x_] = 1 + x*(g[x]^3/6 + g[x^2]*g[x]/2 + g[x^3]/3) + O[x]^n // Normal, {n}]; g[x]];
T[n_, k_] := Module[{t = G[n], g}, t = x*((t^2 + (t /. x -> x^2))/2); g[e_] = (Normal[t + O[x]^Quotient[n, e]] /. x -> x^e) + O[x]^n // Normal; Coefficient[(Sum[EulerPhi[d]*g[d]^(k/d), {d, Divisors[k]}]/k + If[OddQ[ k], g[1]*g[2]^Quotient[k, 2], (g[1]^2 + g[2])*g[2]^(k/2-1)/2])/2, x, n]];
Table[T[n, k], {n, 3, 13}, {k, 3, n}] // Flatten (* Jean-François Alcover, Jul 03 2018, after Andrew Howroyd *)

\\ here G is A000598 as series
G(n)={my(g=O(x)); for(n=1, n, g = 1 + x*(g^3/6 + subst(g, x, x^2)*g/2 + subst(g, x, x^3)/3) + O(x^n)); g}
T(n,k)={my(t=G(n)); t=x*(t^2+subst(t, x, x^2))/2; my(g(e)=subst(t + O(x*x^(n\e)), x, x^e) + O(x*x^n)); polcoeff((sumdiv(k, d, eulerphi(d)*g(d)^(k/d))/k + if(k%2, g(1)*g(2)^(k\2), (g(1)^2+g(2))*g(2)^(k/2-1)/2))/2, n)}

A121331 Number of bridged bicyclic skeletons with n carbon atoms (see Parks et al. for precise definition).

Original entry on oeis.org

1, 2, 6, 15, 39, 99, 258, 671, 1762, 4657, 12372, 33036, 88590, 238483, 644045, 1744542, 4737341, 12894158, 35165994, 96083192, 262951511, 720685274, 1977846334, 5434588909, 14949284828, 41163690109, 113451949753, 312955174089, 863965424349, 2386874582238

Offset: 5

N. J. A. Sloane, Aug 27 2006

nonn

Equivalently, the number of connected graphs on n unlabeled nodes with exactly 2 cycles of the same even length joined along half their length and all nodes having degree at most 4. The resulting graph will have three equal length cycles. - Andrew Howroyd, May 25 2018

			From _Andrew Howroyd_, May 25 2018: (Start)
Illustration of graphs for n=5 and n=6:
    o          o--o       o
   /|\        /|\        /|\
  o o o      o o o      o o o--o
   \|/        \|/        \|/
    o          o          o
.
Illustration of graphs for n=7:
    o--o      o--o--o   o--o        o        o          o   o
   /|\       /|\       /|\         /|\      /|\        /|\ /
  o o o     o o o     o o o--o    o o o    o o o--o   o o o
   \|/       \|/       \|/       / \|/ \    \|/   |    \|/ \
    o--o      o         o       o   o   o    o    o     o   o
(End)

Andrew Howroyd, Table of n, a(n) for n = 5..200
Camden A. Parks and James B. Hendrickson, Enumeration of monocyclic and bicyclic carbon skeletons, J. Chem. Inf. Comput. Sci., vol. 31, 334-339 (1991).

Cf. A121158, A121162, A121165, A305132.

G[n_] := Module[{g}, g[] = 0; Do[g[x] = 1 + x*(g[x]^3/6 + g[x^2]*g[x]/2 + g[x^3]/3) + O[x]^n // Normal, {n}]; g[x]];
C1[n_] := Sum[(d1^(3*k)+3*d1^k*d2^k + 2*d3^k), {k, 1, Quotient[n, 3]}]/12;
C2[n_] := Sum[(d1^Mod[k, 2]*d2^Quotient[k, 2])^3 + 3*d1^Mod[k, 2]* d2^(Quotient[k, 2] + k) + 2*d3^Mod[k, 2]*d6^Quotient[k, 2], {k, 1, Quotient[n, 3]}]/12;
seq[n_] := Module[{s, d, g}, s = G[n]; d = x*(s^2 + (s /. x -> x^2))/2; g[p_, e_] := Normal[(p+O[x]^(Quotient[n, e]+1))] /. x :> x^e; g[s, 1]^2* (C1[n-2] /. Thread[{d1, d2, d3} :> {g[d, 1], g[d, 2], g[d, 3]}]) + g[s, 2]*(C2[n-2] /. Thread[{d1, d2, d3, d6} :> {g[d, 1], g[d, 2], g[d, 3], g[d, 6]}]) + O[x]^n] // CoefficientList[#, x]& // Drop[#, 3]&;
seq[33] (* Jean-François Alcover, Sep 08 2019, after Andrew Howroyd *)

\\ here G is A000598 as series
G(n)={my(g=O(x)); for(n=1, n, g = 1 + x*(g^3/6 + subst(g, x, x^2)*g/2 + subst(g, x, x^3)/3) + O(x^n)); g}
C1(n)={sum(k=1, n\3, (d1^(3*k) + 3*d1^k*d2^k + 2*d3^k))/12}
C2(n)={sum(k=1, n\3, (d1^(k%2)*d2^(k\2))^3 + 3*d1^(k%2)*d2^(k\2+k) + 2*d3^(k%2)*d6^(k\2))/12}
seq(n)={my(s=G(n)); my(d=x*(s^2+subst(s, x, x^2))/2); my(g(p,e)=subst(p + O(x*x^(n\e)), x, x^e)); Vec(O(x^n/x) + g(s,1)^2*substvec(C1(n-2),[d1,d2,d3],[g(d,1), g(d,2), g(d,3)]) + g(s,2)*substvec(C2(n-2), [d1,d2,d3,d6], [g(d,1), g(d,2), g(d,3), g(d,6)]))} \\ Andrew Howroyd, May 25 2018

a(n) ~ c * d^n / sqrt(n), where d = 1/A261340 = 2.815460033176150746526616778..., c = 0.0064202170754... . - Vaclav Kotesovec, Sep 08 2019

Corrected by Franklin T. Adams-Watters and T. D. Noe, Oct 25 2006

a(24) corrected and terms a(26) and beyond from Andrew Howroyd, May 25 2018

A301344 Regular triangle where T(n,k) is the number of semi-binary rooted trees with n nodes and k leaves.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 4, 1, 0, 0, 1, 6, 4, 0, 0, 0, 1, 9, 11, 2, 0, 0, 0, 1, 12, 24, 9, 0, 0, 0, 0, 1, 16, 46, 32, 3, 0, 0, 0, 0, 1, 20, 80, 86, 20, 0, 0, 0, 0, 0, 1, 25, 130, 203, 86, 6, 0, 0, 0, 0, 0, 1, 30, 200, 423, 283, 46, 0, 0, 0, 0, 0, 0, 1, 36, 295, 816, 786, 234, 11, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Mar 19 2018

A rooted tree is semi-binary if all outdegrees are <= 2. The number of semi-binary trees with n nodes is equal to the number of binary trees with n+1 leaves; see A001190.

			Triangle begins:
1
1   0
1   1   0
1   2   0   0
1   4   1   0   0
1   6   4   0   0   0
1   9  11   2   0   0   0
1  12  24   9   0   0   0   0
1  16  46  32   3   0   0   0   0
1  20  80  86  20   0   0   0   0   0
1  25 130 203  86   6   0   0   0   0   0
The T(6,3) = 4 semi-binary rooted trees: ((o(oo))), (o((oo))), (o(o(o))), ((o)(oo)).

Cf. A000081, A000598, A001190, A001678, A003238, A004111, A055277, A111299, A273873, A290689, A291636, A292050, A298118, A298204, A298422, A298426, A301342, A301343, A301345.

rbt[n_]:=rbt[n]=If[n===1,{{}},Join@@Function[c,Union[Sort/@Tuples[rbt/@c]]]/@Select[IntegerPartitions[n-1],Length[#]<=2&]];
Table[Length[Select[rbt[n],Count[#,{},{-2}]===k&]],{n,15},{k,n}]

A002094 Number of unlabeled connected loop-less graphs on n nodes containing exactly one cycle (of length at least 2) and with all nodes of degree <= 4.

Original entry on oeis.org

0, 1, 2, 5, 10, 25, 56, 139, 338, 852, 2145, 5513, 14196, 36962, 96641, 254279, 671640, 1781840, 4742295, 12662282, 33898923, 90981264, 244720490, 659591378, 1781048728, 4817420360, 13050525328, 35405239155, 96180222540, 261603173201, 712364210543

Offset: 1

N. J. A. Sloane

nonn

A pair of parallel edges is permitted and is regarded as a cycle of length 2.
The original definition in A Handbook of Integer Sequences (1973) based on Schiff (1875) was simply "Alcohols". - N. J. A. Sloane, Mar 22 2018
Schiff used an now outdated terminology and did not use the term "alcohols", but in German "zweiwerthige Kohlenwasserstoffe C_{n}H_{2n} ..." and later "... deren je zwei verfuegbare Affinitaeten ... durch Alkoholradikale befriedigt sind.", translated "bivalent hydrocarbons ... whose free valences ... are covered by alcohol radicals". At that time the meaning of "alcohol radical" was different from modern terminology, now meaning an -OH group, but in Schiff's terminology another C_{x}H{y} hydrocarbon group was meant. The organic compounds that are described by the graphs of this sequence in modern chemical terminology are the acyclic alkenes, with exactly one double carbon-to-carbon bond, and the monocyclic cycloalkanes (see Wikipedia links). - Hugo Pfoertner, Mar 29 2018

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 = 1..400
R. J. Mathar, Illustration for graphs up to 6 carbons, 2018
Richard J. Mathar, Counting Connected Graphs without Overlapping Cycles, arXiv:1808.06264 [math.CO], 2018.
Hugo Schiff, Zur Statistik chemischer Verbindungen, Berichte der Deutschen Chemischen Gesellschaft, Vol. 8, pp. 1542-1547, 1875.
Hugo Schiff, Zur Statistik chemischer Verbindungen, Berichte der Deutschen Chemischen Gesellschaft, Vol. 8, pp. 1542-1547, 1875. [Annotated scanned copy]
Wikipedia, Alkene. Those with exactly one double carbon-to-carbon bond are covered by this sequence, the simplest being ethylene C_{2}H_{4}.
Wikipedia, Cycloalkane. The simplest alicyclic compounds, which are the monocyclic saturated hydrocarbons with formula C_{n}H_{2n}, are covered by this sequence, the first example being cyclopropane C_{3}H_{6}.

Cf. A000294, A000598, A000602, A000625, A000642, A001429 (unbound degrees), A068051.

# cycle index of cyclic group C_n
cycC_n := proc(n::integer,a)
    local d ;
    add(numtheory[phi](d)*a[d]^(n/d),d=numtheory[divisors](n)) ;
    %/n ;
end proc:
# cycle index of dihedral group
cyD_n := proc(n::integer,a)
    cycC_n(n,a)/2 ;
    if type(n,'odd') then
        %+ a[1]*a[2]^((n-1)/2)/2 ;
    else
        %+ ( a[1]^2*a[2]^((n-2)/2) +a[2]^(n/2) )/4 ;
    end if;
end proc:
a000642 := [
    1, 1, 2, 3, 7, 14, 32, 72, 171, 405, 989, 2426, 6045, 15167, 38422, 97925,
    251275, 648061, 1679869, 4372872, 11428365, 29972078, 78859809, 208094977,
    550603722, 1460457242, 3882682803, 10344102122, 27612603765, 73844151259,
    197818389539, 530775701520, 1426284383289] ;
g := [add(a000642[i]*x^i,i=1..nops(a000642)) ];
for i from 2 to nops(a000642) do
    g := [op(g), subs(x=x^i,g[1]) ] ;
end do:
Nmax := nops(a000642) :
G := 0 ;
for c from 2 to Nmax do
    f := cyD_n(c,g) ;
    G := G+ taylor(f,x=0,Nmax) ;
end do:
taylor(G,x=0,Nmax) ;
gfun[seriestolist](%) ; # R. J. Mathar, Mar 17 2018

terms = 31;
cycC[n_, a_] := Sum[EulerPhi[d] a[[d]]^(n/d), {d, Divisors[n]}]/n;
cyD[n_, a_] := Module[{cc}, cc = (1/2)cycC[n, a]; If[OddQ[n], (1/2)a[[1]]* a[[2]]^((n-1)/2)+cc, (1/4)(a[[1]]^2 a[[2]]^((n-2)/2) + a[[2]]^(n/2)) + cc]];
B[] = 0; Do[B[x] = Normal[(1/6) x (2 B[x^3] + 3 B[x^2] B[x] + B[x]^3) + O[x]^terms+1], terms];
A[x_] = (1/2) x (B[x^2] + B[x]^2) + O[x]^(terms+2);
a000642 = Rest[CoefficientList[A[x], x]];
g = {Sum[x^i a000642[[i]], {i, 1, terms+1}]};
For[i = 2, i <= Length[a000642], i++, g = Flatten[Append[g, g[[1]] /. x -> x^i]]];
For[G = 0; c = 2, c < terms+1, c++, f = cyD[c, g]; G = Series[f, {x, 0, terms+1}] + G];
Most[Rest[CoefficientList[G, x]]] (* Jean-François Alcover, Mar 26 2020, after R. J. Mathar *)

Let A(x) denote the generating function for A000598 (Number of rooted ternary trees with n nodes), i.e., A(x) = 1+(1/6)*x*(A(x)^3+3*A(x)*A(x^2)+2*A(x^3)), and set B(x)=(A(x)^2+A(x^2))/2. With D_k(x) being the cycle polynomial of the regular k-gon for k>=2, the final generating function is then given by Sum_{k>=2} x^k*D_k(B(x)), which can be evaluated very quickly. - Sascha Kurz, Mar 18 2018

Better definition from R. J. Mathar; terms beyond 852 from R. J. Mathar and Sean A. Irvine, Mar 17 2018

A010372 Number of unrooted quartic trees with n (unlabeled) nodes and possessing a centroid; number of n-carbon alkanes C(n)H(2n +2) with a centroid ignoring stereoisomers.

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 9, 8, 35, 39, 159, 202, 802, 1078, 4347, 6354, 24894, 38157, 148284, 237541, 910726, 1511717, 5731580, 9816092, 36797588, 64658432, 240215803, 431987953, 1590507121, 2917928218, 10660307791, 19910436898

Offset: 1

Paul Zimmermann, N. J. A. Sloane

The degree of each node is <= 4.
A centroid is a node with less than n/2 nodes in each of the incident subtrees, where n is the number of nodes in the tree. If a centroid exists it is unique.
Ignoring stereoisomers means that the children of a node are unordered. They can be permuted in any way and it is still the same tree. See A086194 for the analogous sequence with stereoisomers counted.

F. Harary, Graph Theory, p. 36, for definition of centroid.

A. Cayley, Über die analytischen Figuren, welche in der Mathematik Bäume genannt werden und ihre Anwendung auf die Theorie chemischer Verbindungen, Chem. Ber. 8 (1875), 1056-1059. (Annotated scanned copy)
E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees), J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
Index entries for sequences related to trees

Cf. A010373, A000022, A086194, A000598, A000602.

A000602(n) = a(n) + A010373(n/2) for n even, A000602(n) = a(n) for n odd.

with(combstruct): Alkyl := proc(n) combstruct[count]([ U,{U=Prod(Z,Set(U,card<=3))},unlabeled ],size=n) end:
centeredHC := proc(n) option remember; local f,k,z,f2,f3,f4; f := 1 + add(Alkyl(k)*z^k, k=0..iquo(n-1,2));
f2 := series(subs(z=z^2,f), z, n+1); f3 := series(subs(z=z^3,f), z, n+1); f4 := series(subs(z=z^4,f), z, n+1);
f := series(f*f3/3+f4/4+f2^2/8+f2*f^2/4+f^4/24, z, n+1); coeff(f, z, n-1) end: seq(centeredHC(n), n=1..32);

Description revised by Steve Strand (snstrand(AT)comcast.net), Aug 20 2003

A010373 Number of unrooted quartic trees with 2n (unlabeled) nodes and possessing a bicentroid; number of 2n-carbon alkanes C(2n)H(4n+2) with a bicentroid, ignoring stereoisomers.

Original entry on oeis.org

1, 1, 3, 10, 36, 153, 780, 4005, 22366, 128778, 766941, 4674153, 29180980, 185117661, 1193918545, 7800816871, 51584238201, 344632209090, 2324190638055, 15804057614995, 108277583483391, 746878494484128, 5183852459907628

Offset: 1

Paul Zimmermann, N. J. A. Sloane

The degree of each node is <= 4.
A bicentroid is an edge which connects two subtrees of exactly m/2 nodes, where m is the number of nodes in the tree. If a bicentroid exists it is unique. Clearly trees with an odd number of nodes cannot have a bicentroid.
Ignoring stereoisomers means that the children of a node are unordered. They can be permuted in any way and it is still the same tree. See A086200 for the analogous sequence with stereoisomers counted.

F. Harary, Graph Theory, p. 36, for definition of bicentroid.

Alois P. Heinz, Table of n, a(n) for n = 1..500 (terms n = 1..100 from Vincenzo Librandi)
A. Cayley, Über die analytischen Figuren, welche in der Mathematik Bäume genannt werden und ihre Anwendung auf die Theorie chemischer Verbindungen, Chem. Ber. 8 (1875), 1056-1059. (Annotated scanned copy)
E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees), J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
Index entries for sequences related to trees

A000602(n) = A010372(n) + a(n/2) for n even, A000602(n) = A010372(n) for n odd.

Cf. A000200, A000598.

M[1146] := [ T,{T=Union(Epsilon,U),U=Prod(Z,Set(U,card<=3))},unlabeled ]:
bicenteredHC := proc(n) option remember; if n mod 2<>0 then 0 else binomial(count(M[ 1146 ],size=n/2)+1,2) fi end:

m = 24; a[x_] = Sum[c[k]*x^k, {k, 0, m}]; s[x_] = Series[ 1 + (1/6)*x*(a[x]^3 + 3*a[x]*a[x^2] + 2*a[x^3]) - a[x], {x, 0, m}]; eq = Thread[ CoefficientList[s[x], x] == 0];
Do[so[k] = Solve[eq[[1]], c[k-1]][[1]]; eq = Rest[eq] /. so[k], {k, 1, m+1}]; b = Array[c, m, 0] /. Flatten[ Array[so, m+1] ]; Rest[b*(b+1)/2] (* Jean-François Alcover, Jul 25 2011, after A000598 *)

a(n) = b(n)*(b(n)+1)/2, where b(n) = A000598[ n ].

Description revised by Steve Strand (snstrand(AT)comcast.net), Aug 20 2003

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