A319541 -id:A319541 - cp's OEIS frontend

A001190 Wedderburn-Etherington numbers: unlabeled binary rooted trees (every node has outdegree 0 or 2) with n endpoints (and 2n-1 nodes in all).

0, 1, 1, 1, 2, 3, 6, 11, 23, 46, 98, 207, 451, 983, 2179, 4850, 10905, 24631, 56011, 127912, 293547, 676157, 1563372, 3626149, 8436379, 19680277, 46026618, 107890609, 253450711, 596572387, 1406818759, 3323236238, 7862958391, 18632325319, 44214569100, 105061603969

Offset: 0

N. J. A. Sloane

Also number of n-node binary rooted trees (every node has outdegree <= 2) where root has degree 0 (only for n=1) or 1.
a(n+1) is the number of rooted trees with n nodes where the outdegree of every node is <= 2, see example. These trees are obtained by removing the root of the trees in the comment above. - Joerg Arndt, Jun 29 2014
Number of interpretations of x^n (or number of ways to insert parentheses) when multiplication is commutative but not associative. E.g., a(4) = 2: x(x*x^2) and x^2*x^2. a(5) = 3: (x*x^2)x^2, x(x*x*x^2) and x(x^2*x^2). [If multiplication is non-commutative then the answer is A000108(n-1). - Jianing Song, Apr 29 2022]
Number of ways to place n stars in a single bound stable hierarchical multiple star system; i.e., taking only the configurations from A003214 where all stars are included in single outer parentheses. - Piet Hut, Nov 07 2003
Number of colorations of Kn (complete graph of order n) with n-1 colors such that no triangle is three-colored. Two edge-colorations C1 and C2 of G are isomorphic iff exists an automorphism f (isomorphism between G an G) such that: f sends same-colored edges of C1 on same-colored edges of C2 and f^(-1) sends same-colored edges of C2 on same-colored edges of C1. - Abraham Gutiérrez, Nov 12 2012
For n>1, a(n) is the number of (not necessarily distinct) unordered pairs of free unlabeled trees having a total of n nodes. See the first entry in formula section. - Geoffrey Critzer, Nov 09 2014
Named after the English mathematician Ivor Etherington (1908-1994) and the Scottish mathematician Joseph Wedderburn (1882-1948). - Amiram Eldar, May 29 2021

			G.f. = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 11*x^7 + 23*x^8 + 46*x^9 + 98*x^10 + ...
From _Joerg Arndt_, Jun 29 2014: (Start)
The a(6+1) = 11 rooted trees with 6 nodes as described in the comment are:
:           level sequence       outdegrees (dots for zeros)
:     1:  [ 0 1 2 3 4 5 ]    [ 1 1 1 1 1 . ]
:  O--o--o--o--o--o
:
:     2:  [ 0 1 2 3 4 4 ]    [ 1 1 1 2 . . ]
:  O--o--o--o--o
:           .--o
:
:     3:  [ 0 1 2 3 4 3 ]    [ 1 1 2 1 . . ]
:  O--o--o--o--o
:        .--o
:
:     4:  [ 0 1 2 3 4 2 ]    [ 1 2 1 1 . . ]
:  O--o--o--o--o
:     .--o
:
:     5:  [ 0 1 2 3 4 1 ]    [ 2 1 1 1 . . ]
:  O--o--o--o--o
:  .--o
:
:     6:  [ 0 1 2 3 3 2 ]    [ 1 2 2 . . . ]
:  O--o--o--o
:        .--o
:     .--o
:
:     7:  [ 0 1 2 3 3 1 ]    [ 2 1 2 . . . ]
:  O--o--o--o
:        .--o
:  .--o
:
:     8:  [ 0 1 2 3 2 3 ]    [ 1 2 1 . 1 . ]
:  O--o--o--o
:     .--o--o
:
:     9:  [ 0 1 2 3 2 1 ]    [ 2 2 1 . . . ]
:  O--o--o--o
:     .--o
:  .--o
:
:    10:  [ 0 1 2 3 1 2 ]    [ 2 1 1 . 1 . ]
:  O--o--o--o
:  .--o--o
:
:    11:  [ 0 1 2 2 1 2 ]    [ 2 2 . . 1 . ]
:  O--o--o
:     .--o
:  .--o--o
:
(End)

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Index entries for "core" sequences
Index entries for sequences related to rooted trees
Index entries for sequences related to trees
Index entries for sequences related to parenthesizing

Cf. A000108, A001699, A002658, A003214, A006894, A006961, A088325.
Cf. A086317, A086318, A240943.
Cf. A292553, A292554, A292555, A292556.
Column k=2 of A292085 and of A299038.
Column k=1 of A319539 and of A319541.

A001190 := proc(n) option remember; local s,k; if n<=1 then RETURN(n); elif n <=3 then RETURN(1); else s := 0; if n mod 2 = 0 then s := A001190(n/2)*(A001190(n/2)+1)/2; for k from 1 to n/2-1 do s := s+A001190(k)*A001190(n-k); od; RETURN(s); else for k from 1 to (n-1)/2 do s := s+A001190(k)*A001190(n-k); od; RETURN(s); fi; fi; end;
N := 40: G001190 := add(A001190(n)*x^n,n=0..N);
spec := [S,{S=Union(Z,Prod(Z,Set(S,card=2)))},unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
# alternative Maple program:
a:= proc(n) option remember; `if`(n<2, n, `if`(n::odd, 0,
      (t-> t*(1-t)/2)(a(n/2)))+add(a(i)*a(n-i), i=1..n/2))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 28 2017

terms = 35; A[] = 0; Do[A[x] = x + (1/2)*(A[x]^2 + A[x^2]) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Jean-François Alcover, Jul 22 2011, updated Jan 10 2018 *)
a[n_?OddQ] := a[n] = Sum[a[k]*a[n-k], {k, 1, (n-1)/2}]; a[n_?EvenQ] := a[n] = Sum[a[k]*a[n-k], {k, 1, n/2-1}] + (1/2)*a[n/2]*(1+a[n/2]); a[0]=0; a[1]=1; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Jun 13 2012, after recurrence formula *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Nest[ 1 - Sqrt[1 - 2 x - (# /. x -> x^2)] &, 0, BitLength @ n], {x, 0, n}]]; (* Michael Somos, Apr 25 2013 *)

{a(n) = local(A, m); if( n<0, 0, m=1; A = O(x); while( m<=n, m*=2; A = 1 - sqrt(1 - 2*x - subst(A, x, x^2))); polcoeff(A, n))}; /* Michael Somos, Sep 06 2003 */

{a(n) = local(A); if( n<4, n>0, A = vector(n, i, 1); for( i=4, n, A[i] = sum( j=1, (i-1)\2, A[j] * A[i-j]) + if( i%2, 0, A[i/2] * (A[i/2] + 1)/2)); A[n])}; /* Michael Somos, Mar 25 2006 */

from functools import lru_cache
@lru_cache(maxsize=None)
def A001190(n):
    if n <= 1: return n
    m = n//2 + n % 2
    return sum(A001190(i+1)*A001190(n-1-i) for i in range(m-1)) + (1 - n % 2)*A001190(m)*(A001190(m)+1)//2 # Chai Wah Wu, Jan 14 2022

G.f. satisfies A(x) = x + (1/2)*(A(x)^2 + A(x^2)) [de Bruijn and Klarner].
G.f. also satisfies A(x) = 1 - sqrt(1 - 2*x - A(x^2)). - Michael Somos, Sep 06 2003
a(2n-1) = a(1)a(2n-2) + a(2)a(2n-3) + ... + a(n-1)a(n), a(2n) = a(1)a(2n-1) + a(2)a(2n-2) + ... + a(n-1)a(n+1) + a(n)(a(n)+1)/2.
Given g.f. A(x), then B(x) = -1 + A(x) satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = (u^2 + v)^2 + 2*(v^2 + w). - Michael Somos, Oct 22 2006
The radius of convergence of the g.f. is A240943 = 1/A086317 ~ 0.4026975... - Jean-François Alcover, Jul 28 2014, after Steven R. Finch.
a(n) ~ A086318 * A086317^(n-1) / n^(3/2). - Vaclav Kotesovec, Apr 19 2016

A319539 Array read by antidiagonals: T(n,k) is the number of binary rooted trees with n leaves of k colors and all non-leaf nodes having out-degree 2.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 6, 6, 2, 5, 10, 18, 18, 3, 6, 15, 40, 75, 54, 6, 7, 21, 75, 215, 333, 183, 11, 8, 28, 126, 495, 1260, 1620, 636, 23, 9, 36, 196, 987, 3600, 8010, 8208, 2316, 46, 10, 45, 288, 1778, 8568, 28275, 53240, 43188, 8610, 98, 11, 55, 405, 2970, 17934, 80136, 232500, 366680, 232947, 32763, 207

Offset: 1

Andrew Howroyd, Sep 22 2018

Not all k colors need to be used. The total number of nodes will be 2n-1.

See table 2.1 in the Johnson reference.

			Array begins:
===========================================================
n\k|  1    2      3       4        5        6         7
---+-------------------------------------------------------
1  |  1    2      3       4        5        6         7 ...
2  |  1    3      6      10       15       21        28 ...
3  |  1    6     18      40       75      126       196 ...
4  |  2   18     75     215      495      987      1778 ...
5  |  3   54    333    1260     3600     8568     17934 ...
6  |  6  183   1620    8010    28275    80136    194628 ...
7  | 11  636   8208   53240   232500   785106   2213036 ...
8  | 23 2316  43188  366680  1979385  7960638  26037431 ...
9  | 46 8610 232947 2590420 17287050 82804806 314260765 ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Virginia Perkins Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. South Carolina, 2012.

Columns 1..5 are A001190, A083563, A220816, A220817, A220818.
Main diagonal is A319580.
Cf. A241555, A319254, A319541.

A:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,
      (t-> t*(1-t)/2)(A(n/2, k)))+add(A(i, k)*A(n-i, k), i=1..n/2))
    end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Sep 23 2018

A[n_, k_] := A[n, k] = If[n<2, k*n, If[OddQ[n], 0, (#*(1-#)/2&)[A[n/2, k]]] + Sum[A[i, k]*A[n-i, k], {i, 1, n/2}]];
Table[A[n, d-n+1], {d, 1, 12}, {n, 1, d}] // Flatten (* Jean-François Alcover, Sep 02 2019, after Alois P. Heinz *)

\\ here R(n,k) gives k-th column as a vector.
R(n,k)={my(v=vector(n)); v[1]=k; for(n=2, n, v[n]=sum(j=1, (n-1)\2, v[j]*v[n-j]) + if(n%2, 0, binomial(v[n/2]+1, 2))); v}
{my(T=Mat(vector(8, k, R(8, k)~))); for(n=1, #T~, print(T[n,]))}

T(1,k) = k.
T(n,k) = (1/2)*([n mod 2 == 0]*T(n/2,k) + Sum_{j=1..n-1} T(j,k)*T(n-j,k)).
G.f. of k-th column R(x) satisfies R(k) = k*x + (R(x)^2 + R(x^2))/2.

A319376 Triangle read by rows: T(n,k) is the number of lone-child-avoiding rooted trees with n leaves of exactly k colors.

Original entry on oeis.org

1, 1, 1, 2, 6, 4, 5, 30, 51, 26, 12, 146, 474, 576, 236, 33, 719, 3950, 8572, 8060, 2752, 90, 3590, 31464, 108416, 175380, 134136, 39208, 261, 18283, 245916, 1262732, 3124650, 4014348, 2584568, 660032, 766, 94648, 1908858, 14047288, 49885320, 95715728, 101799712, 56555904, 12818912

Offset: 1

Andrew Howroyd, Sep 17 2018

Lone-child-avoiding rooted trees are also called planted series-reduced trees in some other sequences.

			Triangle begins:
    1;
    1,     1;
    2,     6,      4;
    5,    30,     51,      26;
   12,   146,    474,     576,     236;
   33,   719,   3950,    8572,    8060,   2752;
   90,  3590,  31464,  108416,  175380,  134136,   39208;
  261, 18283, 245916, 1262732, 3124650, 4014348, 2584568, 660032;
  ...
From _Gus Wiseman_, Dec 31 2020: (Start)
The 12 trees counted by row n = 3:
  (111)    (112)    (123)
  (1(11))  (122)    (1(23))
           (1(12))  (2(13))
           (1(22))  (3(12))
           (2(11))
           (2(12))
(End)

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

Columns k=1..2 are A000669, A319377.
Main diagonal is A000311.
Row sums are A316651.
Cf. A141610, A242249, A255517, A256064, A256068, A319254, A319541, A326396.
The unlabeled version, counting inequivalent leaf-colorings of lone-child-avoiding rooted trees, is A330465.
Lone-child-avoiding rooted trees are counted by A001678 (shifted left once).
Labeled lone-child-avoiding rooted trees are counted by A060356.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Cf. A000014, A000081, A000169, A005804, A206429, A330951.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(A(i, k)+j-1, j)*b(n-i*j, i-1, k), j=0..n/i)))
    end:
A:= (n, k)-> `if`(n<2, n*k, b(n, n-1, k)):
T:= (n, k)-> add(A(n, k-j)*(-1)^j*binomial(k, j), j=0..k-1):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Sep 18 2018

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[A[i, k] + j - 1, j] b[n - i j, i - 1, k], {j, 0, n/i}]]];
A[n_, k_] := If[n < 2, n k, b[n, n - 1, k]];
T[n_, k_] := Sum[(-1)^(k - i)*Binomial[k, i]*A[n, i], {i, 1, k}];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 24 2019, after Alois P. Heinz *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
mtot[m_]:=Prepend[Join@@Table[Tuples[mtot/@p],{p,Select[mps[m],1Gus Wiseman, Dec 31 2020 *)

\\ here R(n,k) is k-th column of A319254.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(v=[k]); for(n=2, n, v=concat(v, EulerT(concat(v, [0]))[n])); v}
M(n)={my(v=vector(n, k, R(n,k)~)); Mat(vector(n, k, sum(i=1, k, (-1)^(k-i)*binomial(k,i)*v[i])))}
{my(T=M(10)); for(n=1, #T~, print(T[n, ][1..n]))}

T(n,k) = Sum_{i=1..k} (-1)^(k-i)*binomial(k,i)*A319254(n,i).

Sum_{k=1..n} k * T(n,k) = A326396(n). - Alois P. Heinz, Sep 11 2019

A339650 Triangle read by rows: T(n,k) is the number of trees with n leaves of exactly k colors and all non-leaf nodes having degree 3.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 6, 3, 0, 1, 10, 30, 36, 15, 0, 2, 27, 140, 310, 300, 105, 0, 2, 74, 663, 2376, 3990, 3150, 945, 0, 4, 226, 3186, 17304, 44850, 59805, 39690, 10395, 0, 6, 710, 15642, 123508, 462735, 925890, 1018710, 582120, 135135

Offset: 0

Andrew Howroyd, Dec 14 2020

See table 4.2 in the Johnson reference.

			Triangle begins:
  1;
  0, 1;
  0, 1,   1;
  0, 1,   2,    1;
  0, 1,   4,    6,     3;
  0, 1,  10,   30,    36,    15;
  0, 2,  27,  140,   310,   300,   105;
  0, 2,  74,  663,  2376,  3990,  3150,   945;
  0, 4, 226, 3186, 17304, 44850, 59805, 39690, 10395;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Virginia Perkins Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. South Carolina, 2012.

Columns k=1..4 are A129860, A220829, A220830, A220831.
Main diagonal is A001147(n-2) for n >= 2.
Row sums are A339651.
Cf. A319541 (rooted), A339649, A339780.

\\ here U(n,k) is column k of A339649 as a vector.
R(n, k)={my(v=vector(n)); v[1]=k; for(n=2, n, v[n]=sum(j=1, (n-1)\2, v[j]*v[n-j]) + if(n%2, 0, binomial(v[n/2]+1, 2))); v}
U(n, k)={my(g=x*Ser(R(n, k))); Vec(1 + g + (subst(g + O(x*x^(n\3)), x, x^3) - g^3)/3)}
M(n, m=n)={my(v=vector(m+1, k, U(n, k-1)~)); Mat(vector(m+1, k, k--; sum(i=0, k, (-1)^(k-i)*binomial(k, i)*v[1+i])))}
{my(T=M(10)); for(n=1, #T~, print(T[n, ][1..n]))}

T(n,k) = Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*A339649(n,i).

A319590 Number of binary rooted trees with n leaves spanning an initial interval of positive integers and all non-leaf nodes having out-degree 2.

Original entry on oeis.org

1, 2, 8, 58, 576, 7440, 117628, 2201014, 47552012, 1164812674, 31898271660, 965666303078, 32022547868872, 1154362247246714, 44945574393963472, 1879720975031634318, 84039891496643620196, 3999886612000379135606, 201919706444252727224852, 10775953237291840618917900

Offset: 1

Andrew Howroyd, Sep 23 2018

nonn

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

Row sums of A319541.

Cf. A316651, A319539.

b:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,
      (t-> t*(1-t)/2)(b(n/2, k)))+add(b(i, k)*b(n-i, k), i=1..n/2))
    end:
a:= n-> add(add((-1)^i*binomial(k, i)*b(n, k-i), i=0..k), k=0..n):
seq(a(n), n=1..23);  # Alois P. Heinz, Sep 07 2019

b[n_, k_] := b[n, k] = If[n < 2, k n, If[OddQ[n], 0, Function[t, t(1 - t)/2][b[n/2, k]]] + Sum[b[i, k] b[n - i, k], {i, 1, n/2}]];
a[n_] := Sum[Sum[(-1)^i Binomial[k, i] b[n, k - i], {i, 0, k}], {k, 0, n}];
Array[a, 23] (* Jean-François Alcover, Apr 10 2020, after Alois P. Heinz *)

\\ here R(n, k) is k-th column of A319539 as a vector.
R(n, k)={my(v=vector(n)); v[1]=k; for(n=2, n, v[n]=sum(j=1, (n-1)\2, v[j]*v[n-j]) + if(n%2, 0, binomial(v[n/2]+1, 2))); v}
seq(n)={sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) )}

A220819 Number of rooted binary leaf-multilabeled trees with n leaves on the label set [2].

Original entry on oeis.org

0, 1, 4, 14, 48, 171, 614, 2270, 8518, 32567, 126168, 495079, 1962752, 7853581, 31672502, 128622480, 525523990, 2158818376, 8911039462, 36941520279, 153740822408, 642085403709, 2690217364606, 11304538078369, 47630350694248, 201181246749072, 851690546714230

Offset: 1

N. J. A. Sloane, Dec 22 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1542 (first 200 terms from Andrew Howroyd)
V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012.

Column k=2 of A319541.

Cf. A001190, A083563.

b:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,
      (t-> t*(1-t)/2)(b(n/2, k)))+add(b(i, k)*b(n-i, k), i=1..n/2))
    end:
a:= n-> b(n, 2)-2*b(n, 1):
seq(a(n), n=1..40);  # Alois P. Heinz, Sep 07 2019

(* b = A083563 *) b[n_] := b[n] = If[n < 2, 2*n, If[OddQ[n], 0, #*(1 - #)/2 &[b[n/2]]]] + Sum[b[i]*b[n - i], {i, 1, n/2}];
(* c = A001190 *) c[n_?OddQ] := c[n] = Sum[c[k]*c[n - k], {k, 1, (n - 1)/2}]; c[n_?EvenQ] := c[n] = Sum[c[k]*c[n - k], {k, 1, n/2 - 1}] + (1/2)*c[n/2]*(1 + c[n/2]); c[0] = 0; c[1] = 1;
a[n_] := b[n] - 2 c[n];
Array[a, 27] (* Jean-François Alcover, Sep 07 2019 *)

a(n) = A083563(n) - 2*A001190(n). - Andrew Howroyd, Sep 23 2018

Terms a(11) and beyond from Andrew Howroyd, Sep 23 2018

A220820 Number of rooted binary leaf-multilabeled trees with n leaves on the label set [3].

Original entry on oeis.org

0, 0, 3, 27, 180, 1089, 6333, 36309, 207255, 1184829, 6799473, 39224568, 227554278, 1327688328, 7789644996, 45944320449, 272329169232, 1621636729257, 9697502473014, 58219671592098, 350791368690516, 2120672113779573, 12859687160772771, 78201772222916649

Offset: 1

N. J. A. Sloane, Dec 22 2012

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012.

Column k=3 of A319541.

b:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,
      (t-> t*(1-t)/2)(b(n/2, k)))+add(b(i, k)*b(n-i, k), i=1..n/2))
    end:
a:= n-> (k-> add((-1)^i*binomial(k, i)*b(n, k-i), i=0..k))(3):
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 07 2019

b[n_, k_] := b[n, k] = If[n < 2, k n, If[OddQ[n], 0, Function[t, t(1 - t)/2 ][b[n/2, k]]] + Sum[b[i, k] b[n - i, k], {i, 1, n/2}]];
a[n_] := Function[k, Sum[(-1)^i Binomial[k, i] b[n, k - i], {i, 0, k}]][3];
Array[a, 30] (* Jean-François Alcover, Apr 08 2020, after Alois P. Heinz *)

Terms a(11) and beyond from Andrew Howroyd, Sep 23 2018

A220821 Number of rooted binary leaf-multilabeled trees with n leaves on the label set [4].

Original entry on oeis.org

0, 0, 0, 15, 240, 2604, 24180, 207732, 1710108, 13739550, 108853512, 855732465, 6700902804, 52395480996, 409733313444, 3207687963129, 25155951725808, 197703130100532, 1557413160706764, 12298597436673711, 97359729090421320, 772615510913274126, 6145842794363133324

Offset: 1

N. J. A. Sloane, Dec 22 2012

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012.

Column k=4 of A319541.

b:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,
      (t-> t*(1-t)/2)(b(n/2, k)))+add(b(i, k)*b(n-i, k), i=1..n/2))
    end:
a:= n-> (k-> add((-1)^i*binomial(k, i)*b(n, k-i), i=0..k))(4):
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 07 2019

A[n_, k_] := A[n, k] = If[n < 2, k n, If[OddQ[n], 0, (# (1 - #)/2)&[A[n/2, k]]] + Sum[A[i, k] A[n - i, k], {i, 1, n/2}]];
T[n_, k_] := Sum[(-1)^i Binomial[k, i] A[n, k - i], {i, 0, k}];
a[n_] := T[n, 4];
Array[a, 23] (* Jean-François Alcover, Sep 02 2019, after Alois P. Heinz in A319541 *)

Terms a(11) and beyond from Andrew Howroyd, Sep 23 2018

A220822 Number of rooted binary leaf-multilabeled trees with n leaves on the label set [5].

Original entry on oeis.org

0, 0, 0, 0, 105, 2625, 42075, 554820, 6578550, 73169250, 781319370, 8122058850, 82922497890, 836339477160, 8366154130425, 83235403604220, 825247074255735, 8165187992777430, 80704316324506860, 797435573602269015, 7881226621071221625, 77939438593071188565

Offset: 1

N. J. A. Sloane, Dec 22 2012

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012.

Column 5 of A319541.

b:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,
      (t-> t*(1-t)/2)(b(n/2, k)))+add(b(i, k)*b(n-i, k), i=1..n/2))
    end:
a:= n-> (k-> add((-1)^i*binomial(k, i)*b(n, k-i), i=0..k))(5):
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 07 2019

b[n_, k_] := b[n, k] = If[n < 2, k n, If[OddQ[n], 0, Function[t, t (1 - t)/2][b[n/2, k]]] + Sum[b[i, k] b[n - i, k], {i, 1, n/2}]];
a[n_] := Function[k, Sum[(-1)^i Binomial[k, i] b[n, k - i], {i, 0, k}]][5];
Array[a, 30] (* Jean-François Alcover, Apr 08 2020, after Alois P. Heinz *)

Terms a(11) and beyond from Andrew Howroyd, Sep 23 2018

A319580 Number of binary rooted trees with n leaves of n colors and all non-leaf nodes having out-degree 2.

Original entry on oeis.org

1, 3, 18, 215, 3600, 80136, 2213036, 73068543, 2806959015, 123002168300, 6055381161852, 330885794632536, 19872950226273053, 1301261803764756855, 92259974680854975000, 7041606755629152575055, 575638367425376279620662, 50180725346542105445190603

Offset: 1

Andrew Howroyd, Sep 23 2018

nonn

Not all of the n colors need to be used.

Alois P. Heinz, Table of n, a(n) for n = 1..352
V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012.

Main diagonal of A319539.

Cf. A319369, A319541.

A:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,
      (t-> t*(1-t)/2)(A(n/2, k)))+add(A(i, k)*A(n-i, k), i=1..n/2))
    end:
a:= n-> A(n$2):
seq(a(n), n=1..20);  # Alois P. Heinz, Sep 23 2018

A[n_, k_] := A[n, k] = If[n < 2, k n, If[OddQ[n], 0, Function[t, t(1-t) / 2][A[n/2, k]]] + Sum[A[i, k] A[n - i, k], {i, 1, n/2}]];
a[n_] := A[n, n];
Array[a, 20] (* Jean-François Alcover, Apr 10 2020, after Alois P. Heinz *)

a(n)={my(v=vector(n)); v[1]=n; for(n=2, n, v[n]=sum(j=1, (n-1)\2, v[j]*v[n-j]) + if(n%2, 0, binomial(v[n/2]+1, 2))); v[n]} \\ Andrew Howroyd, Sep 23 2018

cp's OEIS Frontend

A001190 Wedderburn-Etherington numbers: unlabeled binary rooted trees (every node has outdegree 0 or 2) with n endpoints (and 2n-1 nodes in all).

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A319539 Array read by antidiagonals: T(n,k) is the number of binary rooted trees with n leaves of k colors and all non-leaf nodes having out-degree 2.

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A319376 Triangle read by rows: T(n,k) is the number of lone-child-avoiding rooted trees with n leaves of exactly k colors.

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A339650 Triangle read by rows: T(n,k) is the number of trees with n leaves of exactly k colors and all non-leaf nodes having degree 3.

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A319590 Number of binary rooted trees with n leaves spanning an initial interval of positive integers and all non-leaf nodes having out-degree 2.

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A220819 Number of rooted binary leaf-multilabeled trees with n leaves on the label set [2].

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A220820 Number of rooted binary leaf-multilabeled trees with n leaves on the label set [3].

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