A298118 -id:A298118 - cp's OEIS frontend

A000598 Number of rooted ternary trees with n nodes; number of n-carbon alkyl radicals C(n)H(2n+1) ignoring stereoisomers.

1, 1, 1, 2, 4, 8, 17, 39, 89, 211, 507, 1238, 3057, 7639, 19241, 48865, 124906, 321198, 830219, 2156010, 5622109, 14715813, 38649152, 101821927, 269010485, 712566567, 1891993344, 5034704828, 13425117806, 35866550869, 95991365288, 257332864506, 690928354105

Offset: 0

N. J. A. Sloane

Number of unlabeled rooted trees in which each node has out-degree <= 3.
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 A000625 for the analogous sequence with stereoisomers counted.
In alkanes every carbon has valence exactly 4 and every hydrogen has valence exactly 1. But the trees considered here are just the carbon "skeletons" (with all the hydrogen atoms stripped off) so now each carbon bonds to 1 to 4 other carbons. The out-degree is then <= 3.
Other descriptions of this sequence: quartic planted trees with n nodes; ternary rooted trees with n nodes and height at most 3.
The number of aliphatic amino acids with n carbon atoms in the side chain, and no rings or double bonds, has the same growth as this sequence. - Konrad Gruetzmann, Aug 13 2012

			From _Joerg Arndt_, Feb 25 2017: (Start)
The a(5) = 8 rooted trees with 5 nodes and out-degrees <= 3 are:
:         level sequence    out-degrees (dots for zeros)
:     1:  [ 0 1 2 3 4 ]    [ 1 1 1 1 . ]
:  O--o--o--o--o
:
:     2:  [ 0 1 2 3 3 ]    [ 1 1 2 . . ]
:  O--o--o--o
:        .--o
:
:     3:  [ 0 1 2 3 2 ]    [ 1 2 1 . . ]
:  O--o--o--o
:     .--o
:
:     4:  [ 0 1 2 3 1 ]    [ 2 1 1 . . ]
:  O--o--o--o
:  .--o
:
:     5:  [ 0 1 2 2 2 ]    [ 1 3 . . . ]
:  O--o--o
:     .--o
:     .--o
:
:     6:  [ 0 1 2 2 1 ]    [ 2 2 . . . ]
:  O--o--o
:     .--o
:  .--o
:
:     7:  [ 0 1 2 1 2 ]    [ 2 1 . 1 . ]
:  O--o--o
:  .--o--o
:
:     8:  [ 0 1 2 1 1 ]    [ 3 1 . . . ]
:  O--o--o
:  .--o
:  .--o
(End)

N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, p. 62 (quoting Cayley, who is wrong).
A. Cayley, On the mathematical theory of isomers, Phil. Mag. vol. 67 (1874), 444-447 (a(6) is wrong).
J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
R. A. Fisher, Contributions to Mathematical Statistics, Wiley, 1950, 41.397.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 529.
Handbook of Combinatorics, North-Holland '95, p. 1963.
Knop, Mueller, Szymanski and Trinajstich, Computer generation of certain classes of molecules.
D. Perry, The number of structural isomers ..., J. Amer. Chem. Soc. 54 (1932), 2918-2920.
G. Polya, Mathematical and Plausible Reasoning, Vol. 1 Prob. 4 pp. 85; 233.
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 = 0..1000 (first 200 terms from N. J. A. Sloane)
Jean-François Alcover, Mathematica program translated from N. J. A. Sloane's Maple program for A000022, A000200, A000598, A000602, A000678.
A. T. Balaban, J. W. Kennedy and L. V. Quintas, The number of alkanes having n carbons and a longest chain of length d, J. Chem. Education, 65 (1988), 304-313.
A. Cayley, On the mathematical theory of isomers, Phil. Mag. vol. 67 (1874), 444-447 (a(6) is wrong). (Annotated scanned copy)
Frederic Chyzak, Enumerating alcohols and other classes of chemical molecules.
Maximilian Fichtner, K. Voigt, and S. Schuster, The tip and hidden part of the iceberg: Proteinogenic and non-proteinogenic aliphatic amino acids, Biochimica et Biophysica Acta (BBA)-General, 2016, Volume 1861, Issue 1, Part A, January 2017, pp. 3258-3269.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 478.
K. Grützmann, S. Böcker, and S. Schuster, Combinatorics of aliphatic amino acids, Naturwissenschaften, Vol. 98, No. 1, 79-86, 2011.
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 structurally isomeric alcohols of the methanol series, J. Amer. Chem. Soc., 53 (8) (1931), 3042-3045. (Annotated scanned copy)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1.
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
Camden A. Parks and James B. Hendrickson, Enumeration of monocyclic and bicyclic carbon skeletons, J. Chem. Inf. Comput. Sci., vol. 31, 334-339 (1991).
D. Perry, The number of structural isomers of certain homologs of methane and methanol, J. Amer. Chem. Soc. 54 (1932), 2918-2920. [Gives a(60) correctly.] (Annotated scanned copy)
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443; Table I, line 2.
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443; Table I, line 2. (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. 20, Eq. (G); p. 27, Eq. 2.1.
Marko Riedel, Chris Grossack, Counting trees with a degree constraint, Math StackExchange.
R. W. Robinson, F. Harary and A. T. Balaban, Numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (3) (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)
Hugo Schiff, Zur Statistik chemischer Verbindungen, Berichte der Deutschen Chemischen Gesellschaft, Vol. 8, pp. 1542-1547, 1875. [Annotated scanned copy]
N. J. A. Sloane, Maple program and first 60 terms for A000022, A000200, A000598, A000602, A000678.
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.
Wikipedia, Polya's enumeration theorem.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A000599, A000600, A000602, A000625, A000628, A000678, A010372, A010373, A086194, A086200, A261340.
Cf. A292553, A292554, A292555, A292556.
Cf. A000081, A001190, A014591, A032305, A295461, A298118, A298120, A298204, A298422, A298426.
Column k=3 of A299038.

N := 45; G000598 := 0: i := 0: while i<(N+1) do G000598 := series(1+z*(G000598^3/6+subs(z=z^2,G000598)*G000598/2+subs(z=z^3,G000598)/3)+O(z^(N+1)),z,N+1): t[ i ] := G000598: i := i+1: od: A000598 := n->coeff(G000598,z,n);
# Another Maple program for g.f. G000598:
G000598 := 1; f := proc(n) global G000598; coeff(series(1+(1/6)*x*(G000598^3+3*G000598*subs(x=x^2,G000598)+2*subs(x=x^3,G000598)),x, n+1),x,n); end; for n from 1 to 50 do G000598 := series(G000598+f(n)*x^n,x,n+1); od; G000598;
spec := [S, {Z=Atom, S=Union(Z, Prod(Z, Set(S, card=3)))}, unlabeled]: [seq(combstruct[count](spec, size=n), n=0..20)];

m = 45; Clear[f]; f[1, x_] := 1+x; f[n_, x_] := f[n, x] = Expand[1+x*(f[n-1, x]^3/6 + f[n-1, x^2]*f[n-1, x]/2 + f[n-1, x^3]/3)][[1 ;; n]]; Do[f[n, x], {n, 2, m}]; CoefficientList[f[m, x], x]
(* second program (after N. J. A. Sloane): *)
m = 45; gf[] = 0; Do[gf[z] = 1 + z*(gf[z]^3/6 + gf[z^2]*gf[z]/2 + gf[z^3]/3) + O[z]^m // Normal, m]; CoefficientList[gf[z], z]  (* Jean-François Alcover, Sep 23 2014, updated Jan 11 2018 *)
b[0, i_, t_, k_] = 1; m = 3; (* m = maximum children *)
b[n_,i_,t_,k_]:= b[n,i,t,k]= 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]}]];
Join[{1},Table[b[n-1, n-1, m, m], {n, 1, 35}]] (* Robert A. Russell, Dec 27 2022 *)

seq(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)); Vec(g)} \\ Andrew Howroyd, May 22 2018

def seq(n):
    B = PolynomialRing(QQ, 't', n+1);t = B.gens()
    R. = B[[]]
    T = sum([t[i] * z^i for i in range(1,n+1)]) + O(z^(n+1))
    lhs, rhs = T, 1 + z/6 * (T(z)^3 + 3*T(z)*T(z^2) + 2*T(z^3))
    I = B.ideal([lhs.coefficients()[i] - rhs.coefficients()[i] for i in range(n)])
    return [I.reduce(t[i]) for i in range(1,n+1)]
seq(33) # Chris Grossack, Mar 31 2025

G.f. A(x) satisfies A(x) = 1 + (1/6)*x*(A(x)^3 + 3*A(x)*A(x^2) + 2*A(x^3)).

a(n) ~ c * d^n / n^(3/2), where d = 1/A261340 = 2.8154600331761507465266167782426995425365065396907..., c = 0.517875906458893536993162356992854345458168348098... . - Vaclav Kotesovec, Aug 15 2015

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

A298422 Number of rooted trees with n nodes in which all positive outdegrees are the same.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 5, 2, 6, 4, 9, 2, 20, 2, 26, 12, 53, 2, 120, 2, 223, 43, 454, 2, 1100, 11, 2182, 215, 4902, 2, 11446, 2, 24744, 1242, 56014, 58, 131258, 2, 293550, 7643, 676928, 2, 1582686, 2, 3627780, 49155, 8436382, 2, 19809464, 50, 46027323, 321202

Offset: 1

Gus Wiseman, Jan 19 2018

nonn

Row sums of A298426.

			The a(9) = 6 trees: ((((((((o)))))))), (o(o(o(oo)))), (o((oo)(oo))), ((oo)(o(oo))), (ooo(oooo)), (oooooooo).

Alois P. Heinz, Table of n, a(n) for n = 1..200

Cf. A000005, A000081, A000598, A001190, A001678, A003238, A004111, A008864, A032305, A067538, A111299, A124343, A143773, A289078, A289079, A295461, A298118, A298204, A298423, A298424, A298426.

srut[n_]:=srut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[srut/@c]]]/@Select[IntegerPartitions[n-1],Function[ptn,And@@(Divisible[#-1,Length[ptn]]&/@ptn)]],SameQ@@Length/@Cases[#,{},{0,Infinity}]&]];
Table[srut[n]//Length,{n,20}]

a(n) = 2 <=> n in {A008864}. - Alois P. Heinz, Jan 20 2018

a(44)-a(52) from Alois P. Heinz, Jan 20 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}]

A301342 Regular triangle where T(n,k) is the number of rooted identity trees with n nodes and k leaves.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 4, 1, 0, 0, 0, 1, 6, 5, 0, 0, 0, 0, 1, 9, 13, 2, 0, 0, 0, 0, 1, 12, 28, 11, 0, 0, 0, 0, 0, 1, 16, 53, 40, 3, 0, 0, 0, 0, 0, 1, 20, 91, 109, 26, 0, 0, 0, 0, 0, 0, 1, 25, 146, 254, 116, 6, 0, 0, 0, 0, 0, 0, 1, 30, 223, 524, 387, 61, 0, 0, 0, 0, 0, 0, 0, 1, 36

Offset: 1

Gus Wiseman, Mar 19 2018

			Triangle begins:
1
1   0
1   0   0
1   1   0   0
1   2   0   0   0
1   4   1   0   0   0
1   6   5   0   0   0   0
1   9  13   2   0   0   0   0
1  12  28  11   0   0   0   0   0
1  16  53  40   3   0   0   0   0   0
1  20  91 109  26   0   0   0   0   0   0
1  25 146 254 116   6   0   0   0   0   0   0
1  30 223 524 387  61   0   0   0   0   0   0   0
The T(6,2) = 4 rooted identity trees: (((o(o)))), ((o((o)))), (o(((o)))), ((o)((o))).

A version with the zeroes removed is A055327.

Cf. A000081, A001190, A003238, A004111, A032305, A055277, A273873, A276625, A277098, A290689, A298118, A298422, A298426, A301343, A301344, A301345.

irut[n_]:=irut[n]=If[n===1,{{}},Join@@Function[c,Select[Union[Sort/@Tuples[irut/@c]],UnsameQ@@#&]]/@IntegerPartitions[n-1]];
Table[Length[Select[irut[n],Count[#,{},{-2}]===k&]],{n,8},{k,n}]

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 *)

A300439 Number of odd enriched p-trees of weight n (all outdegrees are odd).

Original entry on oeis.org

1, 1, 2, 2, 5, 7, 18, 29, 75, 132, 332, 651, 1580, 3268, 7961, 16966, 40709, 89851, 215461, 484064, 1159568, 2641812, 6337448, 14622880, 35051341, 81609747, 196326305, 459909847, 1107083238, 2611592457, 6299122736, 14926657167, 36069213786, 85809507332

Offset: 1

Gus Wiseman, Mar 05 2018

nonn

An odd enriched p-tree of weight n > 0 is either a single node of weight n, or a finite odd-length sequence of at least 3 odd enriched p-trees whose weights are weakly decreasing and sum to n.

			The a(6) = 7 odd enriched p-trees: 6, (411), (321), (222), ((111)21), ((211)11), (21111).

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000009, A027193, A063834, A078408, A196545, A273873, A289501, A294079, A298118, A299202, A299203, A300300, A300301, A300436, A300440.

f[n_]:=f[n]=1+Sum[Times@@f/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&OddQ[Length[#]]&]}];
Array[f,40]

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x*x^n)) - 1/prod(k=1, n-1, 1 + v[k]*x^k + O(x*x^n)), n)/2); v} \\ Andrew Howroyd, Aug 26 2018

A300436 Number of odd p-trees of weight n (all proper terminal subtrees have odd weight).

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 5, 12, 13, 35, 37, 98, 107, 304, 336, 927, 1037, 3010, 3367, 9585, 10924, 32126, 36438, 105589, 121045, 359691, 412789, 1211214, 1398168, 4188930, 4831708, 14315544, 16636297, 50079792, 58084208, 173370663, 202101971, 611487744, 712709423

Offset: 1

Gus Wiseman, Mar 05 2018

nonn

An odd p-tree of weight n > 0 is either a single node (if n = 1) or a finite sequence of at least 3 odd p-trees whose weights are weakly decreasing odd numbers summing to n.

			The a(7) = 5 odd p-trees: ((ooo)(ooo)o), (((ooo)oo)oo), ((ooooo)oo), ((ooo)oooo), (ooooooo).

Cf. A000009, A027193, A063834, A078408, A196545, A279374, A279785, A289501, A298118, A299202, A299203, A300300, A300301, A300355, A300439, A300440.

b[n_]:=b[n]=If[n>1,0,1]+Sum[Times@@b/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&And@@OddQ/@#&]}];
Table[b[n],{n,40}]

O.g.f: x + Product_{n odd} 1/(1 - a(n)*x^n) - Sum_{n odd} a(n)*x^n. - Gus Wiseman, Aug 27 2018

Name corrected by Gus Wiseman, Aug 27 2018

A358334 Number of twice-partitions of n into odd-length partitions.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 25, 43, 77, 137, 241, 410, 720, 1209, 2073, 3498, 5883, 9768, 16413, 26978, 44741, 73460, 120462, 196066, 320389, 518118, 839325, 1353283, 2178764, 3490105, 5597982, 8922963, 14228404, 22609823, 35875313, 56756240, 89761600, 141410896, 222675765

Offset: 0

Gus Wiseman, Dec 01 2022

nonn

A twice-partition of n (A063834) is a sequence of integer partitions, one of each part of an integer partition of n.

			The a(0) = 1 through a(5) = 13 twice-partitions:
  ()  ((1))  ((2))     ((3))        ((4))           ((5))
             ((1)(1))  ((111))      ((211))         ((221))
                       ((2)(1))     ((2)(2))        ((311))
                       ((1)(1)(1))  ((3)(1))        ((3)(2))
                                    ((111)(1))      ((4)(1))
                                    ((2)(1)(1))     ((11111))
                                    ((1)(1)(1)(1))  ((111)(2))
                                                    ((211)(1))
                                                    ((2)(2)(1))
                                                    ((3)(1)(1))
                                                    ((111)(1)(1))
                                                    ((2)(1)(1)(1))
                                                    ((1)(1)(1)(1)(1))

Andrew Howroyd, Table of n, a(n) for n = 0..1000

For multiset partitions of integer partitions: A356932, ranked by A356935.
For odd length instead of lengths we have A358824.
For odd sums instead of lengths we have A358825.
For odd sums also we have A358827.
For odd length also we have A358834.
A000041 counts integer partitions.
A027193 counts odd-length partitions, ranked by A026424.
A055922 counts partitions with odd multiplicities, also odd parts A117958.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
Cf. A000219, A001970, A072233, A078408, A270995, A279374, A298118, A300300, A300301, A300647, A302243.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],OddQ[Times@@Length/@#]&]],{n,0,10}]

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
R(u,y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
seq(n) = {my(u=Vec(P(n,1)-P(n,-1))/2); Vec(R(u, 1), -(n+1))} \\ Andrew Howroyd, Dec 30 2022

G.f.: 1/Product_{k>=1} (1 - A027193(k)*x^k). - Andrew Howroyd, Dec 30 2022

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2022

cp's OEIS Frontend

A000598 Number of rooted ternary trees with n nodes; number of n-carbon alkyl radicals C(n)H(2n+1) ignoring stereoisomers.

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A298422 Number of rooted trees with n nodes in which all positive outdegrees are the same.

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

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A301342 Regular triangle where T(n,k) is the number of rooted identity trees with n nodes and k leaves.

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

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A300439 Number of odd enriched p-trees of weight n (all outdegrees are odd).

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A300436 Number of odd p-trees of weight n (all proper terminal subtrees have odd weight).

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A358334 Number of twice-partitions of n into odd-length partitions.