A000014 -id:A000014 - cp's OEIS frontend

A358376 Numbers k such that the k-th standard ordered rooted tree is lone-child-avoiding (counted by A005043).

1, 4, 8, 16, 18, 25, 32, 36, 50, 57, 64, 72, 100, 114, 121, 128, 137, 144, 200, 228, 242, 249, 256, 258, 274, 281, 288, 385, 393, 400, 456, 484, 498, 505, 512, 516, 548, 562, 569, 576, 770, 786, 793, 800, 897, 905, 912, 968, 996, 1010, 1017, 1024, 1032, 1096

Offset: 1

Gus Wiseman, Nov 14 2022

nonn

We define the n-th standard ordered rooted tree to be obtained by taking the (n-1)-th composition in standard order (graded reverse-lexicographic, A066099) as root and replacing each part with its own standard ordered rooted tree. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.

			The initial terms and their corresponding trees:
    1: o
    4: (oo)
    8: (ooo)
   16: (oooo)
   18: ((oo)o)
   25: (o(oo))
   32: (ooooo)
   36: ((oo)oo)
   50: (o(oo)o)
   57: (oo(oo))
   64: (oooooo)
   72: ((oo)ooo)
  100: (o(oo)oo)
  114: (oo(oo)o)
  121: (ooo(oo))
  128: (ooooooo)
  137: ((oo)(oo))
  144: ((oo)oooo)
  200: (o(oo)ooo)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

These trees are counted by A005043.
The series-reduced case appears to be counted by A284778.
The unordered version is A291636, counted by A001678.
A000081 counts unlabeled rooted trees, ranked by A358378.
A358371 and A358372 count leaves and nodes in standard ordered rooted trees.
A358374 ranks ordered identity trees, counted by A032027.
A358375 ranks ordered binary trees, counted by A126120.
Cf. A000014, A001263, A001679, A004249, A061775, A063895, A187306, A331489, A331490, A331934, A358373, A358377.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
Select[Range[100],FreeQ[srt[#],[_]?(Length[#]==1&)]&]

A059123 Number of homeomorphically irreducible rooted trees (also known as series-reduced rooted trees, or rooted trees without nodes of degree 2) with n >= 1 nodes.

Original entry on oeis.org

0, 1, 1, 0, 2, 2, 4, 6, 12, 20, 39, 71, 137, 261, 511, 995, 1974, 3915, 7841, 15749, 31835, 64540, 131453, 268498, 550324, 1130899, 2330381, 4813031, 9963288, 20665781, 42947715, 89410092, 186447559, 389397778, 814447067, 1705775653

Offset: 0

Wolfdieter Lang, Jan 09 2001

Essentially the same as A001679. - Eric W. Weisstein, Mar 25 2022

			G.f. = x + x^2 + 2*x^4 + 2*x^5 + 4*x^6 + 6*x^7 + 12*x^8 + 20*x^9 + ...

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 62, Eq. (3.3.9).

N. J. A. Sloane, Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014)
P. J. Cameron, Some treelike objects, Quart. J. Math. Oxford, 38 (1987), 155-183.
N. J. A. Sloane, Illustration of initial terms
Index entries for sequences related to trees
Index entries for sequences related to rooted trees

Cf. A001679.
Cf. A000055 (trees by nodes), A000014 (homeomorphically irreducible trees by nodes), A000669 (homeomorphically irreducible planted trees by leaves), A000081 (rooted trees by nodes).
Cf. A246403.

with(powseries): with(combstruct): n := 30: Order := n+3: sys := {B = Prod(C,Z), S = Set(B,1 <= card), C = Union(Z,S)}:
G001678 := (convert(gfseries(sys,unlabeled,x)[S(x)], polynom)) * x^2: G0temp := G001678 + x^2:
G059123 := G0temp / x + G0temp - (G0temp^2+eval(G0temp,x=x^2))/(2*x): A059123 := 0,seq(coeff(G059123,x^i),i=1..n); # Ulrich Schimke (ulrschimke(AT)aol.com)

terms = 36; (* F = G001678 *) F[] = 0; Do[F[x] = (x^2/(1 + x))*Exp[Sum[ F[x^k]/(k*x^k), {k, 1, j}]] + O[x]^j // Normal, {j, 1, terms + 1}];
G[x_] = 1 + ((1 + x)/x)*F[x] - (F[x]^2 + F[x^2])/(2*x) + O[x]^terms;
CoefficientList[G[x] - 1, x] (* Jean-François Alcover, May 25 2012, updated Jan 12 2018 *)

{a(n) = local(A); if( n<3, n>0, A = x / (1 - x^2) + x * O(x^n); for(k=3, n-1, A /= (1 - x^k + x * O(x^n))^polcoeff(A, k)); polcoeff( (1 + x) * A - x * (A^2 + subst(A, x, x^2)) / 2, n))}; /* Michael Somos, Jun 13 2014 */

G.f.: 1 + ((1+x)/x)*f(x) - (f(x)^2+f(x^2))/(2*x) where 1+f(x) is g.f. for A001678 (homeomorphically irreducible planted trees by nodes).
a(n) = A001679(n) if n>0. - Michael Somos, Jun 13 2014
a(n) ~ c * d^n / n^(3/2), where d = A246403 = 2.18946198566085056388702757711... and c = 0.421301852869924921096502830935802411658488216342994235732491571594804013... - Vaclav Kotesovec, Jun 26 2014

A254382 Number of rooted labeled trees on n nodes such that every nonroot node is the child of a branching node or of the root.

Original entry on oeis.org

0, 1, 2, 3, 16, 85, 696, 6349, 72080, 918873, 13484080, 219335281, 3962458248, 78203547877, 1680235050872, 38958029188485, 970681471597216, 25847378934429361, 732794687650764000, 22032916968153975769, 700360446794528578520

Offset: 0

Geoffrey Critzer, Jan 29 2015

nonn

Here, a branching node is a node with at least two children.
In other words, a(n) is the number of labeled rooted trees on n nodes such that the path from every node towards the root reaches a branching node (or the root) in one step.
Also labeled rooted trees that are lone-child-avoiding except possibly for the root. The unlabeled version is A198518. - Gus Wiseman, Jan 22 2020

			a(5) = 85:
...0................0...............0-o...
...|.............../ \............ /|\....
...o..............o   o...........o o o...
../|\............/ \   ...................
.o o o..........o   o   ..................
These trees have 20 + 60 + 5 = 85 labelings.
From _Gus Wiseman_, Jan 22 2020: (Start)
The a(1) = 1 through a(4) = 16 trees (in the format root[branches]) are:
  1  1[2]  1[2,3]  1[2,3,4]
     2[1]  2[1,3]  1[2[3,4]]
           3[1,2]  1[3[2,4]]
                   1[4[2,3]]
                   2[1,3,4]
                   2[1[3,4]]
                   2[3[1,4]]
                   2[4[1,3]]
                   3[1,2,4]
                   3[1[2,4]]
                   3[2[1,4]]
                   3[4[1,2]]
                   4[1,2,3]
                   4[1[2,3]]
                   4[2[1,3]]
                   4[3[1,2]]
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..200
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Cf. A231797, A052318 (condition is applied only to leaf nodes).
The unlabeled version is A198518
The non-planted case is A060356.
Labeled rooted trees are A000169.
Lone-child-avoiding rooted trees are A001678(n + 1).
Labeled topologically series-reduced rooted trees are A060313.
Labeled lone-child-avoiding unrooted trees are A108919.
Cf. A000014, A000669, A001679, A005512, A291636, A331488, A331578.

nn = 20; b = 1 + Sum[nn = n; n! Coefficient[Series[(Exp[x] - x)^n, {x, 0, nn}], x^n]*x^n/n!, {n,1, nn}]; c = Sum[a[n] x^n/n!, {n, 0, nn}]; sol = SolveAlways[b == Series[1/(1 - (c - x)), {x, 0, nn}], x]; Flatten[Table[a[n], {n, 0, nn}] /. sol]
nn = 30; CoefficientList[Series[1+x-1/Sum[SeriesCoefficient[(E^x-x)^n,{x,0,n}]*x^n,{n,0,nn}],{x,0,nn}],x] * Range[0,nn]! (* Vaclav Kotesovec, Jan 30 2015 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
lrt[set_]:=If[Length[set]==0,{},Join@@Table[Apply[root,#]&/@Join@@Table[Tuples[lrt/@stn],{stn,sps[DeleteCases[set,root]]}],{root,set}]];
Table[Length[Select[lrt[Range[n]],FreeQ[Z@@#,Integer[]]&]],{n,6}] (* Gus Wiseman, Jan 22 2020 *)

E.g.f.: A(x) satisfies 1/(1 - (A(x) - x)) = B(x) where B(x) is the e.g.f. for A231797.

a(n) ~ (1-exp(-1))^(n-1/2) * n^(n-1). - Vaclav Kotesovec, Jan 30 2015

A331489 Matula-Goebel numbers of topologically series-reduced rooted trees.

Original entry on oeis.org

1, 2, 7, 8, 16, 19, 28, 32, 43, 53, 56, 64, 76, 98, 107, 112, 128, 131, 152, 163, 172, 196, 212, 224, 227, 256, 263, 266, 304, 311, 343, 344, 383, 392, 424, 428, 443, 448, 512, 521, 524, 532, 577, 602, 608, 613, 652, 686, 688, 719, 722, 742, 751, 784, 848, 856

Offset: 1

Gus Wiseman, Jan 20 2020

nonn

We say that a rooted tree is topologically series-reduced if no vertex (including the root) has degree 2.

The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of its branches. This gives a bijective correspondence between positive integers and unlabeled rooted trees.

			The sequence of all topologically series-reduced rooted trees together with their Matula-Goebel numbers begins:
    1: o
    2: (o)
    7: ((oo))
    8: (ooo)
   16: (oooo)
   19: ((ooo))
   28: (oo(oo))
   32: (ooooo)
   43: ((o(oo)))
   53: ((oooo))
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   98: (o(oo)(oo))
  107: ((oo(oo)))
  112: (oooo(oo))
  128: (ooooooo)
  131: ((ooooo))
  152: (ooo(ooo))
  163: ((o(ooo)))

Eric Weisstein's World of Mathematics, Series-reduced tree.
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Unlabeled rooted trees are counted by A000081.
Topologically series-reduced trees are counted by A000014.
Topologically series-reduced rooted trees are counted by A001679.
Labeled topologically series-reduced trees are counted by A005512.
Labeled topologically series-reduced rooted trees are counted by A060313.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Cf. A000669, A001678, A007097, A007821, A060356, A061775, A109082, A109129, A196050, A254382, A276625, A330943, A331490.

nn=1000;
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
srQ[n_]:=Or[n==1,With[{m=primeMS[n]},And[Length[m]>1,And@@srQ/@m]]];
Select[Range[nn],PrimeOmega[#]!=2&&And@@srQ/@primeMS[#]&]

A331578 Number of labeled series-reduced rooted trees with n vertices and more than two branches of the root.

Original entry on oeis.org

0, 0, 0, 4, 5, 186, 847, 17928, 166833, 3196630, 45667391, 925287276, 17407857337, 393376875906, 8989368580935, 229332484742416, 6094576250570849, 174924522900914094, 5271210321949744111, 168792243040279327860, 5674164658298121248361, 200870558472768096534490

Offset: 1

Gus Wiseman, Jan 21 2020

nonn

A rooted tree is series-reduced if no vertex (including the root) has degree 2.

Also labeled lone-child-avoiding rooted trees with n vertices and more than two branches, where a rooted tree is lone-child-avoiding if no vertex has exactly one child.

			Non-isomorphic representatives of the a(7) = 847 trees (in the format root[branches]) are:
  1[2,3,4[5,6,7]]
  1[2,3,4,5[6,7]]
  1[2,3,4,5,6,7]

Andrew Howroyd, Table of n, a(n) for n = 1..200

The non-series-reduced version is A331577.
The unlabeled version is A331488.
Lone-child-avoiding rooted trees are counted by A001678.
Topologically series-reduced rooted trees are counted by A001679.
Labeled topologically series-reduced rooted trees are counted by A060313.
Labeled lone-child-avoiding rooted trees are counted by A060356.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Matula-Goebel numbers of series-reduced rooted trees are A331489.
Cf. A000014, A000169, A000669, A005512, A108919, A206429, A331233, A331490.

lrt[set_]:=If[Length[set]==0,{},Join@@Table[Apply[root,#]&/@Join@@Table[Tuples[lrt/@stn],{stn,sps[DeleteCases[set,root]]}],{root,set}]];
Table[Length[Select[lrt[Range[n]],Length[#]>2&&FreeQ[#,[]]&]],{n,6}]

a(n) = {if(n<=1, 0, sum(k=1, n, (-1)^(n-k)*k^(k-2)*n*(n-2)!*binomial(n-1,k-1)*(2*k*n - n - k^2)/k!))} \\ Andrew Howroyd, Dec 09 2020

seq(n)={my(w=lambertw(-x/(1+x) + O(x*x^n))); Vec(serlaplace(-x - w - (x/2)*w^2), -n)} \\ Andrew Howroyd, Dec 09 2020

From Andrew Howroyd, Dec 09 2020: (Start)
a(n) = A060313(n) - n*A060356(n-1) for n > 1.
a(n) = Sum_{k=1..n} (-1)^(n-k)*k^(k-2)*n*(n-2)!*binomial(n-1,k-1)*(2*k*n - n - k^2)/k! for n > 1.
E.g.f.: -x - LambertW(-x/(1+x)) - (x/2)*LambertW(-x/(1+x))^2.
(End)

Terms a(9) and beyond from Andrew Howroyd, Dec 09 2020

A091967 a(n) is the n-th term of sequence A_n, ignoring the offset, or -1 if A_n has fewer than n terms.

Original entry on oeis.org

0, 2, 1, 0, 2, 3, 0, 6, 6, 4, 44, 1, 180, 42, 16, 1096, 7652, 13781, 8, 24000, 119779, 458561, 152116956851941670912, 1054535, -53, 26, 27, 59, 4806078, 2, 35792568, 3010349, 2387010102192469724605148123694256128, 2, 0, -53, 43, 0, -4097, 173, 37338, 111111111111111111111111111111111111111111, 30402457, 413927966

Offset: 1

Proposed by several people, including Jeff Burch and Michael Joseph Halm

sign

This version ignores the offset of A_n and just counts from the beginning of the terms shown in the OEIS entry.
Thus a(8) = 6 because A_8 begins 1,1,2,2,3,4,5,6,... [even though A_8(8) is really 7].
The value a(n) = -1 could arise in two different ways, but it will be easy to decide which. - N. J. A. Sloane, Nov 27 2016
From M. F. Hasler, Sep 22 2013: (Start)
The value of a(91967) can be chosen at will.
Note that this sequence may change if the initial terms in A_n are altered, which does happen from time to time, usually because of the addition of an initial term.
After a(47), currently unknown, the sequence continues with a(48) = A48(47) = 1497207322929, a(49) = A49(48) = unknown, a(50) = A50(49) = unknown, a(51) = A51(50) = 1125899906842625, a(52)=97, a(53) = -1 (since A000053 has only 29 terms). (End)
a(58) = A000058(57) = 138752...985443 (29334988649136302 digits) is too large to include in the b-file. - Pontus von Brömssen, May 21 2022

			a(1) = 0 since A000001 has offset 0, and begins with A000001(0) = 0.
a(26) = 26 because the 26th term of A000026 = 26.

Pontus von Brömssen, Table of n, a(n) for n = 1..57 (terms n = 1..46 from M. F. Hasler)
OEIS, List of finite sequences, sorted by A-number.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Index entries for sequences whose definition involves A_n (or An).

See A051070, A107357, A102288 for other versions.

Cf. A000001, A000002, A000003, A000004, A000005, A000006, A000007, A000008, A000009, A000010, A000011, A000012, A000013, A000014, A000015, etc.

Corrected and extended by Jud McCranie; further extended by N. J. A. Sloane and E. M. Rains, Dec 08 1998
Corrected and extended by N. J. A. Sloane, May 25 2005
a(26), a(36) and a(42) corrected by M. F. Hasler, Jan 30 2009
a(43) and a(44) added by Daniel Sterman, Nov 27 2016
a(1) corrected by N. J. A. Sloane, Nov 27 2016 at the suggestion of Daniel Sterman
Definition and comments changed by N. J. A. Sloane, Nov 27 2016

A246403 Decimal expansion of a constant related to series-reduced trees.

Original entry on oeis.org

2, 1, 8, 9, 4, 6, 1, 9, 8, 5, 6, 6, 0, 8, 5, 0, 5, 6, 3, 8, 8, 7, 0, 2, 7, 5, 7, 7, 1, 1, 4, 5, 4, 4, 9, 6, 7, 3, 3, 1, 7, 0, 8, 7, 4, 4, 2, 3, 8, 4, 9, 0, 3, 0, 1, 0, 5, 0, 2, 7, 3, 4, 2, 5, 3, 5, 7, 1, 5, 6, 2, 5, 7, 0, 4, 2, 2, 1, 2, 2, 6, 3, 0, 0, 8, 5, 8, 6, 0, 7, 8, 4, 8, 1, 9, 3, 3, 3, 0, 8, 3, 2, 0, 3, 8

Offset: 1

Vaclav Kotesovec, Aug 25 2014

			2.189461985660850563887027577114544967331...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 302 and 561.

Cf. A000014, A001678, A001679, A059123, A244456, A198518.

Equals lim n -> infinity A000014(n)^(1/n).
Equals lim n -> infinity A001678(n)^(1/n).
Equals lim n -> infinity A001679(n)^(1/n).
Equals lim n -> infinity A059123(n)^(1/n).
Equals lim n -> infinity A244456(n)^(1/n).
Equals lim n -> infinity A198518(n)^(1/n).

More terms from Vaclav Kotesovec, Sep 03 2014 and Dec 26 2020

A271205 Number T(m,n) of series-reduced free trees with n nodes of which exactly m >= 3 are leaves, m+1 <= n <= 2m-2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 4, 4, 2, 1, 3, 6, 10, 8, 4, 1, 3, 9, 17, 22, 15, 6, 1, 4, 12, 30, 47, 53, 32, 11, 1, 4, 16, 44, 91, 127, 121, 66, 18, 1, 5, 20, 67, 158, 282, 346, 292, 142, 37, 1, 5, 25, 91, 258, 539, 841, 921, 688, 306, 66, 1, 6, 30, 126, 397, 978, 1804, 2498, 2456, 1662, 672, 135, 1, 6, 36, 163, 588, 1636, 3550, 5856, 7260, 6489, 3978, 1483, 265, 1, 7, 42, 213, 838, 2638, 6495, 12554, 18636, 20946, 17082, 9629, 3316, 552, 1, 8

Offset: 3

Stephan Beyer, Apr 01 2016

The sequence of row sums a(m) = Sum_{n} T(m,n) is A007827.

The sequence of column sums a(n) = Sum_{m} T(m,n) is A000014.

			m\n | 3 4 5 6 7 8 9 10 11 12 13 14 15 16  17  18 19 20
-------------------------------------------------------
3   | . 1 . . . . .  .  .  .  .  .  .  .   .   .  .  .
4   | . . 1 1 . . .  .  .  .  .  .  .  .   .   .  .  .
5   | . . . 1 1 1 .  .  .  .  .  .  .  .   .   .  .  .
6   | . . . . 1 2 2  2  .  .  .  .  .  .   .   .  .  .
7   | . . . . . 1 2  4  4  2  .  .  .  .   .   .  .  .
8   | . . . . . . 1  3  6 10  8  4  .  .   .   .  .  .
9   | . . . . . . .  1  3  9 17 22 15  6   .   .  .  .
10  | . . . . . . .  .  1  4 12 30 47 53  32  11  .  .
11  | . . . . . . .  .  .  1  4 16 44 91 127 121 66 18

Stephan Beyer, Python code to generate the sequence on GitHub Gist

Cf. A000014, A007827.

Transpose of A271362.

A378079 Number of series-reduced noncrossing trees with n edges.

Original entry on oeis.org

1, 1, 0, 4, 5, 33, 91, 408, 1485, 6195, 24838, 103752, 432796, 1834140, 7815900, 33591376, 145197017, 631281591, 2757917260, 12102728740, 53321334381, 235768155073, 1045889996047, 4653534540816, 20761857325000, 92862669150004, 416316199107096, 1870414803490240

Offset: 0

Andrew Howroyd, Nov 21 2024

nonn

			The a(3) = 4 trees are:
    o---o    o---o    o   o    o   o
    | \        / |    | /        \ |
    o   o    o   o    o---o    o---o

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A000014, A001764, A030980, A378078.

seq(n)={my(g=serreverse(x/(1/(1-x)^2 - 2*x) + O(x*x^n))); Vec(1/(1 - g) - g^2)}

G.f.: 1/(1 - g(x)) - g(x)^2 where g(x) is the g.f. of A030980.

A238416 Triangle read by rows: T(n,k) is the number of trees with n vertices having k vertices of degree 2 (n>=2, 0 <= k <= n - 2).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 2, 0, 1, 2, 3, 2, 3, 0, 1, 4, 4, 7, 3, 4, 0, 1, 5, 9, 10, 12, 5, 5, 0, 1, 10, 15, 25, 20, 22, 6, 7, 0, 1, 14, 31, 46, 54, 38, 34, 9, 8, 0, 1, 26, 57, 103, 111, 114, 65, 53, 11, 10, 0, 1, 42, 114, 204, 267, 250, 212, 108, 76, 15, 12, 0, 1, 78, 219, 440, 583, 644, 502, 383, 167, 110, 18, 14, 0, 1

Offset: 2

Emeric Deutsch, Mar 05 2014

Sum of entries in row n is A000055(n) (number of trees with n vertices).
T(n,0) = A000014(n) (= number of series-reduced trees with n vertices).
The author knows of no formula for T(n,k). The entries have been obtained in the following manner, explained for row n = 7. In A235111 we find that the 11 (=A000055(7)) trees with 7 vertices have M-indices 25, 27, 30, 35, 36, 40, 42, 48, 49, 56, and 64 (the M-index of a tree t is the smallest of the Matula numbers of the rooted trees isomorphic, as a tree, to t). Making use of the formula in A182907 for the degree sequence polynomial, from these Matula numbers one obtains that these trees have 5, 3, 3, 3, 2, 2, 1, 1, 1, 0, and 0 degree-2 vertices, respectively; the frequencies of 0, 1, 2, 3, 4, and 5 are 2, 3, 2, 3, 0, and 1, respectively. See the Maple program.

			Row n=4 is T(4,0)=1,T(4,1)=0; T(4,2)=1; indeed, the star S[4] has no degree-2 vertex and the path P[4] has 2 degree-2 vertices.
Triangle starts:
1;
0, 1;
1, 0, 1;
1, 1, 0, 1;
2, 1, 2, 0, 1;
2, 3, 2, 3, 0, 1;
4, 4, 7, 3, 4, 0, 1;
5, 9, 10, 12, 5, 5, 0, 1.

Andrew Howroyd, Table of n, a(n) for n = 2..1226 (rows 2..50)

Cf. A000055, A000014, A182907.

MI := [25, 27, 30, 35, 36, 40, 42, 48, 49, 56, 64]: with(numtheory): g := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 1 elif bigomega(n) = 1 then sort(expand(g(pi(n))+x^bigomega(pi(n))*(x-1)+x)) else sort(expand(g(r(n))+g(s(n))-x^bigomega(r(n))-x^bigomega(s(n))+x^bigomega(n))) end if end proc: a := proc (n) options operator, arrow: coeff(g(n), x, 2) end proc: G := add(x^a(MI[q]), q = 1 .. 11): seq(coeff(G, x, j), j = 0 .. 5);

EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
T(n)={my(u=[1]); for(n=2, n, u=concat([1], EulerMT(u) + (y-1)*u)); my(r=x*Ser(u), v=Vec(-x + r*(1 + x*(1-y)) + (substvec(r,[x,y],[x^2,y^2])*(1 - x*(1-y)) - r^2*(1 + x*(1-y)))/2)); [Vecrev(p) | p<-v]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Dec 20 2020

G.f.: -x + R(x,y)*(1 + x*(1-y)) + (R(x^2,y^2)*(1 - x*(1-y)) - R(x,y)^2*(1 + x*(1-y)))/2 where R(x,y) satisfies R(x,y) = x*(R(x,y)*(y-1) + exp(Sum_{k>0} R(x^k,y^k)/k)). - Andrew Howroyd, Dec 20 2020

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