A000598 -id:A000598 - cp's OEIS frontend

A298118 Number of unlabeled rooted trees with n nodes in which all positive outdegrees are odd.

1, 1, 1, 2, 3, 6, 11, 21, 40, 80, 159, 322, 657, 1356, 2816, 5896, 12407, 26267, 55861, 119331, 255878, 550665, 1188786, 2574006, 5588177, 12162141, 26529873, 57993624, 127020653, 278716336, 612617523, 1348680531, 2973564157, 6565313455, 14514675376

Offset: 1

Gus Wiseman, Jan 12 2018

nonn

			The a(6) = 6 trees: (((((o))))), (((ooo))), ((oo(o))), (oo((o))), (o(o)(o)), (ooooo).

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000081, A000598, A003238, A004111, A027193, A032305, A067659, A290689, A291443, A297791, A298120.

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

a(n) ~ c * d^n / n^(3/2), where d = 2.30984417428419893876754252289588812511559... and c = 0.5598122522173731208680575003383895445787... - Vaclav Kotesovec, Jun 04 2019

a(24)-a(35) from Alois P. Heinz, Jan 12 2018

A298426 Regular triangle where T(n,k) is number of k-ary rooted trees with n nodes.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 2, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 3, 0, 1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 6, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 11, 4, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 23, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gus Wiseman, Jan 19 2018

Row sums are A298422.

			Triangle begins:
1
0  1
0  1  1
0  1  0  1
0  1  1  0  1
0  1  0  0  0  1
0  1  2  1  0  0  1
0  1  0  0  0  0  0  1
0  1  3  0  1  0  0  0  1
0  1  0  2  0  0  0  0  0  1
0  1  6  0  0  1  0  0  0  0  1
0  1  0  0  0  0  0  0  0  0  0  1
0  1  11 4  2  0  1  0  0  0  0  0  1
0  1  0  0  0  0  0  0  0  0  0  0  0  1
0  1  23 0  0  0  0  1  0  0  0  0  0  0  1
0  1  0  8  0  2  0  0  0  0  0  0  0  0  0  1

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

Cf. A000005, A000081, A000598, A001190, A001678, A003238, A004111, A067538, A143773, A289078, A289079, A295461, A298118, A298422, A298423, A298424.

nn=16;
arut[n_,k_]:=If[n===1,{{}},Join@@Function[c,Union[Sort/@Tuples[arut[#,k]&/@c]]]/@Select[IntegerPartitions[n-1],Length[#]===k&]]
Table[arut[n,k]//Length,{n,nn},{k,0,n-1}]

A000628 Number of n-node unrooted steric quartic trees; number of n-carbon alkanes C(n)H(2n+2) taking stereoisomers into account.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 11, 24, 55, 136, 345, 900, 2412, 6563, 18127, 50699, 143255, 408429, 1173770, 3396844, 9892302, 28972080, 85289390, 252260276, 749329719, 2234695030, 6688893605, 20089296554, 60526543480, 182896187256, 554188210352, 1683557607211, 5126819371356, 15647855317080, 47862049187447, 146691564302648, 450451875783866, 1385724615285949

Offset: 0

N. J. A. Sloane

Trees are unrooted; nodes are unlabeled and have degree <= 4.
Regarding stereoisomers as different means that only the alternating group A_4 acts at each node, not the full symmetric group S_4. See A000602 for the analogous sequence when stereoisomers are not counted as different.
Has also been described as steric planted trees (paraffins) with n nodes.

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 290.
R. Davies and P. J. Freyd, C_{167}H_{336} is The Smallest Alkane with More Realizable Isomers than the Observable Universe has Particles, Journal of Chemical Education, Vol. 66, 1989, pp. 278-281.
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).

Charles M. Blair and Henry R. Henze, The number of stereoisomeric and non-stereoisomeric paraffin hydrocarbons, J. Amer. Chem. Soc., 54 (1932), 1538-1545.
Charles M. Blair and Henry R. Henze, The number of stereoisomeric and non-stereoisomeric paraffin hydrocarbons, J. Amer. Chem. Soc., 54 (4) (1932), 1538-1545. (Annotated scanned copy)
L. Bytautats and D. J. Klein, Alkane Isomer Combinatorics: Stereostructure enumeration and graph-invariant and molecular-property distributions, J. Chem. Inf. Comput. Sci 39 (1999) 803-818, Table 1.
Lorentz Jäntschi and Lavinia-Lorena Pruteanu, Geometry of C_32 cyclic polyyne and some of its clusters, Carpathian J. Math. (2025) Vol. 41, No. 2, 371-391. See p. 371.
Pierre Leroux and Brahim Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992.
Pierre Leroux and Brahim Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (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. 44.
R. W. Robinson, F. Harary and A. T. Balaban, The numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (1976), 355-361.
R. W. Robinson, F. Harary and A. T. Balaban, Numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (3) (1976), 355-361. (Annotated scanned copy)
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Equals A000626 + A000627.

Cf. A000598, A000602, A000625, A010372, A010373, A086194, A086200.

s[0]:=1:s[1]:=1:for n from 0 to 60 do s[n+1/3]:=0 od:for n from 0 to 60 do s[n+2/3]:=0 od:for n from 0 to 60 do s[n+1/4]:=0 od:for n from 0 to 60 do s[n+1/2]:=0 od:for n from 0 to 60 do s[n+3/4]:=0 od:s[ -1]:=0:for n from 1 to 50 do s[n+1]:=(2*n/3*s[n/3]+sum(j*s[j]*sum(s[k]*s[n-j-k],k=0..n-j),j=1..n))/n od:for n from 0 to 50 do q[n]:=sum(s[i]*s[n-i],i=0..n) od:for n from 0 to 50 do q[n-1/2]:=0 od:for n from 0 to 40 do f:=n->(3*s[n]+2*s[n/2]+q[(n-1)/2]-q[n]+2*sum(s[j]*s[n-3*j-1],j=0..n/3))/4 od:seq(f(n),n=0..38); # the formulas for s[n+1] and f(n) are from eq.(4) and (12), respectively, of the Robinson et al. paper; s[n]=A000625(n), f(n)=A000628(n); q[n] is the convolution of s[n] with itself; # Emeric Deutsch

max = 40; s[0] = s[1] = 1; s[] = 0; For[n=1, n <= max, n++, s[n+1] = (2*n/3*s[n/3] + Sum[j*s[j]*Sum[s[k]*s[n-j-k], {k, 0, n-j}], {j, 1, n}])/n]; For[n=0, n <= max, n++, q[n] = Sum[s[i]*s[n-i], {i, 0, n}]]; For[n=0, n <= max, n++, q[n-1/2]=0]; f[n] := (3*s[n] + 2*s[n/2] + q[(n-1)/2] - q[n] + 2*Sum[s[j]*s[n-3*j-1], {j, 0, n/3}])/4; Table[f[n], {n, 0, max}] (* Jean-François Alcover, Dec 29 2014, after Emeric Deutsch *)

Blair and Henze give recurrence (see the Maple code).

For even n a(n) = A086194(n) + A086200(n/2), for odd n a(n) = A086194(n).

Additional comments from Steve Strand (snstrand(AT)comcast.net), Aug 20 2003

More terms from Emeric Deutsch, May 16 2004

A299038 Number A(n,k) of rooted trees with n nodes where each node has at most k children; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 2, 3, 1, 0, 1, 1, 1, 2, 4, 6, 1, 0, 1, 1, 1, 2, 4, 8, 11, 1, 0, 1, 1, 1, 2, 4, 9, 17, 23, 1, 0, 1, 1, 1, 2, 4, 9, 19, 39, 46, 1, 0, 1, 1, 1, 2, 4, 9, 20, 45, 89, 98, 1, 0, 1, 1, 1, 2, 4, 9, 20, 47, 106, 211, 207, 1, 0

Offset: 0

Alois P. Heinz, Feb 01 2018

			Square array A(n,k) begins:
  1, 1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
  1, 1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
  0, 1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
  0, 1,   2,   2,   2,   2,   2,   2,   2,   2,   2, ...
  0, 1,   3,   4,   4,   4,   4,   4,   4,   4,   4, ...
  0, 1,   6,   8,   9,   9,   9,   9,   9,   9,   9, ...
  0, 1,  11,  17,  19,  20,  20,  20,  20,  20,  20, ...
  0, 1,  23,  39,  45,  47,  48,  48,  48,  48,  48, ...
  0, 1,  46,  89, 106, 112, 114, 115, 115, 115, 115, ...
  0, 1,  98, 211, 260, 277, 283, 285, 286, 286, 286, ...
  0, 1, 207, 507, 643, 693, 710, 716, 718, 719, 719, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=1-11 give: A000012, A001190(n+1), A000598, A036718, A036721, A036722, A182378, A292553, A292554, A292555, A292556.
Main diagonal gives A000081 for n>0.
A(2n,n) gives A299039.
Cf. A244372.

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, k)-> `if`(n=0, 1, b(n-1$2, k$2)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

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_, k_] := If[n == 0, 1, b[n - 1, n - 1, k, k]];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 04 2018, from Maple *)

from sympy import binomial
from sympy.core.cache import cacheit
@cacheit
def b(n, i, t, k): return 1 if n==0 else 0 if i<1 else sum([binomial(b(i-1, i-1, k, k)+j-1, j)*b(n-i*j, i-1, t-j, k) for j in range(min(t, n//i)+1)])
def A(n, k): return 1 if n==0 else b(n-1, n-1, k, k)
for d in range(15): print([A(n, d-n) for n in range(d+1)]) # Indranil Ghosh, Mar 02 2018, after Maple code

A(n,k) = Sum_{i=0..k} A244372(n,i) for n>0, A(0,k) = 1.

A000022 Number of centered hydrocarbons with n atoms.

Original entry on oeis.org

0, 1, 0, 1, 1, 2, 2, 6, 9, 20, 37, 86, 181, 422, 943, 2223, 5225, 12613, 30513, 74883, 184484, 458561, 1145406, 2879870, 7274983, 18471060, 47089144, 120528657, 309576725, 797790928, 2062142876, 5345531935, 13893615154, 36201693122

Offset: 0

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

R. G. Busacker and T. L. Saaty, Finite Graphs and Networks, McGraw-Hill, NY, 1965, p. 201. (They reproduce Cayley's mistakes.)
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.
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, On the mathematical theory of isomers, Phil. Mag. vol. 67 (1874), 444-447.
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)
H. R. Henze and C. M. Blair, The number of structurally isomeric alcohols of the methanol series, J. Amer. Chem. Soc., 53 (8) (1931), 3042-3046.
H. R. Henze and C. M. Blair, The number of isomeric hydrocarbons of the methane series, J. Amer. Chem. Soc., 53 (8) (1931), 3077-3085.
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, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Maple program and first 60 terms for A000022, A000200, A000598, A000602, A000678
Index entries for sequences related to trees

A000022 + A000200 = A000602 for n>0. Cf. A010372.

# We continue from the Maple code in A000678: Unordered 4-tuples of ternary trees with one of height i and others of height at most i-1:
N := 45: i := 1: while i<(N+1) do Tb := t[ i ]-t[ i-1 ]: Ts := t[ i ]-1: Q2 := series(Tb*Ts+O(z^(N+1)),z,200): q2[ i ] := Q2: i := i+1; od: q2[ 0 ] := 0: q[ -1 ] := 0:
for i from 0 to N do c[ i ] := series(q[ i ]-q[ i-1 ]-q2[ i ]+O(z^(N+1)),z,200); od:
# erase height information: i := 'i': cent := series(sum(c[ i ],i=0..N),z,200); G000022 := cent; A000022 := n->coeff(G000022,z,n);
# continued in A000200.

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;
S4[f_,h_,x_] := f[h,x]^4/24 + f[h,x]^2 f[h,x^2]/4 + f[h,x] f[h,x^3]/3 + f[h,x^2]^2/8 + f[h,x^4]/4;
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) + S4[T,h-1,z]z - S4[T,h-2,z]z - (T[h-1,z] - T[h-2,z]) (T[h-1,z]-1),z], n+1], {h,1,n/2}] + PadRight[{0,1}, n+1] (* Robert A. Russell, Sep 15 2018 *)

A000642 a(1)=0; for n>1, a(n) = number of isomeric hydrocarbons of the acetylene series with carbon content n.

Original entry on oeis.org

0, 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

Offset: 1

N. J. A. Sloane

The former definition was "Number of alkyl derivatives of acetylene X^{II} C_n H_{2n+2} with n carbon atoms" with offset 0.

a(n+1) is the number of rooted trees with n nodes and out-degree <= 2 on the root and out-degree <= 3 on all other nodes. See illustration of initial terms. - Washington Bomfim, Nov 28 2020

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).

T. D. Noe, Table of n, a(n) for n = 1..201 (offset adapted by _Georg Fischer_, Jan 31 2019)
Washington Bomfim, Illustration of initial terms
D. D. Coffman, C. M. Blair and H. R. Henze, The number of structurally isomeric hydrocarbons of the acetylene series, J. Amer. Chem. Soc., 55 (1933), 252-253.
D. D. Coffman, C. M. Blair and H. R. Henze, The number of structurally isomeric hydrocarbons of the acetylene series, J. Amer. Chem. Soc., 55 (1933), 252-253. (Annotated scanned copy)
Jean-Loup Faulon, Donald P. Visco Jr., Diana Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
R. J. Mathar, Illustration for graphs up to 6 carbons
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443; line 8 of Table I, "R" of Table IV.
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443; line 8 of Table I, "R" of Table IV. (Annotated scanned copy)
R. C. Read, Some recent results in chemical enumeration, Lect. Notes Math. 303 (1972), 243-259.
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. A000598, A002094.

terms = 32; B[] = 0; Do[B[x] = 1 + (1/6)*x*(B[x]^3 + 3*B[x]*B[x^2] + 2*B[x^3]) + O[x]^terms // Normal, terms];
A[x_] = (1/2)*x*(B[x^2] + B[x]^2) + O[x]^terms;
CoefficientList[A[x], x] (* Jean-François Alcover, Jun 28 2012, updated Jan 10 2018 *)

\\ here G(n) is A000598 as g.f.
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(g=G(n)); Vec(subst(g,x,x^2) + g^2, -(n+1))/2} \\ Andrew Howroyd, Nov 28 2020

G.f.: A(x)=(1/2)*x*(B(x^2)+B(x)^2), where B(x) = g.f. for A000598.

a(n) ~ c * d^n / n^(3/2), where d = 1/A261340 = 2.815460033176... and c = 0.13833565403175156418512996853... - Vaclav Kotesovec, Feb 11 2019

I changed the definition and offset so as to agree with Coffman et al. (1933). - N. J. A. Sloane, Jan 13 2019

A292556 Number of rooted unlabeled trees on n nodes where each node has at most 11 children.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12485, 32970, 87802, 235355, 634771, 1720940, 4688041, 12824394, 35216524, 97039824, 268238379, 743596131, 2066801045, 5758552717, 16080588286, 44997928902, 126160000878, 354349643101, 996946927831

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, A292555.

Column k=11 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, 11$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, 11, 11]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jun 05 2018, after Alois P. Heinz *)

Functional equation of g.f. is T(z) = z + z*Sum_{q=1..11} 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_11)(T(z)).
a(n) = Sum_{j=1..11} A244372(n,j) for n > 0, a(0) = 1. - Alois P. Heinz, Sep 20 2017
Limit_{n->oo} a(n)/a(n+1) = 0.338324339068091181557475416836618315086769320447748735003402... - Robert A. Russell, Feb 11 2023

A298204 Number of unlabeled rooted trees with n nodes in which all outdegrees are either 0, 1, or 3.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 16, 29, 55, 104, 200, 389, 763, 1507, 3002, 6010, 12102, 24484, 49751, 101475, 207723, 426542, 878451, 1813945, 3754918, 7790326, 16196629, 33739335, 70410401, 147187513, 308171861, 646188276, 1356847388, 2852809425, 6005542176

Offset: 1

Gus Wiseman, Jan 14 2018

nonn

			The a(7) = 9 trees: ((((((o)))))), ((((ooo)))), (((oo(o)))), ((oo((o)))), ((o(o)(o))), (oo(((o)))), (oo(ooo)), (o(o)((o))), ((o)(o)(o)).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Gus Wiseman, Illustration of the first 8 terms.

Cf. A000081, A000598, A001190, A001399, A004111, A014591, A032305, A273873, A292050, A295461, A298118, A298205.

b:= proc(n, i, v) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or n `if`(n<2, n, add(b(n-1$2, j), j=[1, 3])):
seq(a(n), n=1..40);  # Alois P. Heinz, Jan 30 2018

multing[n_,k_]:=Binomial[n+k-1,k];
a[n_]:=a[n]=If[n===1,1,Sum[Product[multing[a[x],Count[ptn,x]],{x,Union[ptn]}],{ptn,Select[IntegerPartitions[n-1],MemberQ[{1,3},Length[#]]&]}]];
Table[a[n],{n,40}]
(* Second program: *)
b[n_, i_, v_] := b[n, i, v] = If[n == 0,
     If[v == 0, 1, 0], If[i < 1 || v < 1 || n < v, 0,
     If[n == v, 1, Sum[Binomial[a[i] + j - 1, j]*
     b[n - i*j, i - 1, v - j], {j, 0, Min[n/i, v]}]]]];
a[n_] := If[n < 2, n, Sum[b[n - 1, n - 1, j], {j, {1, 3}}]];
Array[a, 40] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

A292553 Number of rooted unlabeled trees on n nodes where each node has at most 8 children.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 20, 48, 115, 286, 718, 1839, 4757, 12460, 32897, 87592, 234746, 633013, 1715851, 4673320, 12781759, 35093010, 96681705, 267199518, 740580555, 2058042803, 5733101603, 16006590851, 44782679547, 125533577578, 352525803976, 991634575368

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.
Marko Riedel, Maple code for sequences A001190, A000598, A036718, A036721, A036722, A182378, A292553, A292554, A292555, A292556 (FEQ 1).
Marko Riedel, Maple code for sequences A001190, A000598, A036718, A036721, A036722, A182378, A292553, A292554, A292555, A292556 (FEQ 2)
Marko Riedel, Maple code (FEQ 2) optimized for speed.

Cf. A000081, A001190, A000598, A036718, A036721, A036722, A182378, A244372, A292554, A292555, A292556.

Column k=8 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, 8$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, 8, 8]];
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..8} 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_8)(T(z)).
a(n) = Sum_{j=1..8} A244372(n,j) for n>0, a(0) = 1. - Alois P. Heinz, Sep 20 2017
a(n) / a(n+1) ~ 0.338386042364849957035744926227166370702775721795018600630554... - Robert A. Russell, Feb 11 2023

A292554 Number of rooted unlabeled trees on n nodes where each node has at most 9 children.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1841, 4763, 12477, 32947, 87735, 235162, 634212, 1719325, 4683368, 12810871, 35177357, 96926335, 267909285, 742641309, 2064029034, 5750500663, 16057186086, 44929879114, 125962026154, 353773417487, 995269027339

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, A292555, A292556.

Column k=9 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, 9$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, 9, 9]];
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..9} 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_9)(T(z)).
a(n) = Sum_{j=1..9} A244372(n,j) for n>0, a(0) = 1. - Alois P. Heinz, Sep 20 2017
a(n) / a(n+1) ~ 0.338343552789108712866488147828528012266693326385052387884853... - Robert A. Russell, Feb 11 2023

cp's OEIS Frontend

A298118 Number of unlabeled rooted trees with n nodes in which all positive outdegrees are odd.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A298426 Regular triangle where T(n,k) is number of k-ary rooted trees with n nodes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A000628 Number of n-node unrooted steric quartic trees; number of n-carbon alkanes C(n)H(2n+2) taking stereoisomers into account.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A299038 Number A(n,k) of rooted trees with n nodes where each node has at most k children; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A000022 Number of centered hydrocarbons with n atoms.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

A000642 a(1)=0; for n>1, a(n) = number of isomeric hydrocarbons of the acetylene series with carbon content n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A292556 Number of rooted unlabeled trees on n nodes where each node has at most 11 children.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula