A038062 -id:A038062 - cp's OEIS frontend

A038055 Number of n-node rooted trees with nodes of 2 colors.

2, 4, 14, 52, 214, 916, 4116, 18996, 89894, 433196, 2119904, 10503612, 52594476, 265713532, 1352796790, 6933598208, 35747017596, 185260197772, 964585369012, 5043220350012, 26467146038744, 139375369621960, 736229024863276, 3900074570513316, 20714056652990194

Offset: 1

Christian G. Bower, Jan 04 1999

Alois P. Heinz, Table of n, a(n) for n = 1..1000
L. Foissy, Algebraic structures on typed decorated rooted trees, arXiv:1811.07572 [math.RA] (2018).
R. J. Mathar, Topologically distinct sets of non-intersecting circles in the plane, arXiv:1603.00077 [math.CO] (2016), Table 3.
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A000081, A038056-A038062, A271878 (multisets).

Cf. A245870.

spec := [N, {N=Prod(bead,Set(N)), bead=Union(R,B), R=Atom, B=Atom}]; [seq(combstruct[count](spec, size=n), n=1..40)];
# second Maple program:
with(numtheory):
a:= proc(n) option remember; `if`(n<2, 2*n, (add(add(d*
      a(d), d=divisors(j))*a(n-j), j=1..n-1))/(n-1))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, May 11 2014

a[n_] := a[n] = If[n<2, 2*n, (Sum[Sum[d*a[d], {d, Divisors[j]}]*a[n-j], {j, 1, n-1}])/(n-1)]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 25 2015, after Alois P. Heinz *)
a[1] = 2; a[n_] := a[n] = Sum[k a[k] a[n - m k]/(n-1), {k, n}, {m, (n-1)/k}]; Table[a[n], {n, 30}] (* Vladimir Reshetnikov, Aug 12 2016 *)

seq(N) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 2/n * sum(i=1, n, sumdiv(i, d, d*A[d]) * A[n-i+1] ) ); 2*A} \\ Andrew Howroyd, May 12 2018

Shifts left and halves under Euler transform.
a(n) = 2*A000151(n).
a(n) ~ c * d^n / n^(3/2), where d = A245870 = 5.646542616232949712892713516..., c = 0.41572319484583484264330698410170337587070758092051645875080558178621559423... . - Vaclav Kotesovec, Sep 11 2014, updated Dec 26 2020

A038056 Number of trees with n 2-colored nodes.

Original entry on oeis.org

2, 3, 6, 18, 54, 189, 700, 2778, 11486, 49377, 217936, 984818, 4533004, 21198962, 100471862, 481754806, 2333465628, 11404047817, 56177574628, 278710061814, 1391617108600, 6988781174376, 35283390923324, 178990052273558, 912019786448570, 4665994686718118

Offset: 1

Christian G. Bower, Jan 04 1999

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for sequences related to trees

Cf. A000055, A038055-A038062. Row sums of A294783.

G.f.: B(x) - B^2(x)/2 + B(x^2)/2, where B(x) is g.f. for A038055.

A038057 a(n) = 2^n*n^(n-1).

Original entry on oeis.org

2, 8, 72, 1024, 20000, 497664, 15059072, 536870912, 22039921152, 1024000000000, 53119845582848, 3043362286338048, 190857913323364352, 13004222844995895296, 956593800000000000000, 75557863725914323419136

Offset: 1

Christian G. Bower, Jan 04 1999

nonn

Labeled rooted trees with n 2-colored nodes.

Index entries for sequences related to rooted trees

Cf. A000169, A019762 (2*e), A038055-A038062.

Equals 2 * A052746(n).

nn=16;f[x_]:=Sum[a[n]x^n/n!,{n,0,nn}];s=SolveAlways[0==Series[f[x]-2x Exp[f[x]],{x,0,nn}],x];Table[a[n],{n,1,nn}]/.s
(* or *)
nn=16;Drop[Range[0,nn]!CoefficientList[Series[-LambertW[-2x],{x,0,nn}],x],1]
(* or *)
Table[2^n*n^(n-1),{n,1,16}]  (* Geoffrey Critzer, Mar 17 2013 *)

E.g.f.: B(2*x) where B(x) is e.g.f. of A000169.

Limit_{n->oo} a(n+1)/(n*a(n)) = 2*e. - Stefano Spezia, Mar 12 2023

New description from Vladeta Jovovic, Mar 08 2003

A038059 Number of rooted trees with 3-colored nodes.

Original entry on oeis.org

3, 9, 45, 246, 1485, 9432, 62625, 428319, 3000393, 21410436, 155106693, 1137703869, 8432624850, 63060142671, 475196487363, 3604851603690, 27507181503069, 210988219961637, 1625848092941463, 12580709718788622, 97714211996345868, 761528782558088202

Offset: 1

Christian G. Bower, Jan 04 1999

Shifts left and divides by 3 under Euler transform.

L. Foissy, Algebraic structures on typed decorated rooted trees, arXiv:1811.07572 [math.RA] (2018).
R. J. Mathar, Topologically distinct sets of non-intersecting circles in the plane, arXiv:1603.00077 [math.CO] (2016), Table 4.
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A000081, A038055-A038062, A271879 (multisets).

with(numtheory): a:= proc(n) option remember; `if`(n<2, 3*n, (add(add(d*a(d), d=divisors(j)) *a(n-j), j=1..n-1))/ (n-1)) end: seq(a(n), n=1..30); # Alois P. Heinz, Sep 06 2008

a[n_] := a[n] = If[n<2, 3*n, Sum[Sum[d*a[d], {d, Divisors[j]}] *a[n-j], {j, 1, n-1}]/(n-1)]; Array[a, 30] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz *)

a(n) = 3 * A006964(n).

A038058 Number of labeled trees with 2-colored nodes.

Original entry on oeis.org

1, 2, 4, 24, 256, 4000, 82944, 2151296, 67108864, 2448880128, 102400000000, 4829076871168, 253613523861504, 14681377947951104, 928873060356849664, 63772920000000000000, 4722366482869645213696, 375183514207494575620096

Offset: 0

Christian G. Bower, Jan 04 1999

nonn

Robert Israel, Table of n, a(n) for n = 0..321
Index entries for sequences related to trees

Cf. A000272, A038055-A038062.

1, seq(2^n * n^(n-2), n=1..20); # Robert Israel, Nov 02 2014

nn = 17; f[x_] := Sum[n^(n - 2) x^n/n!, {n, 1, nn}];
Range[0, nn]! CoefficientList[Series[f[2 x] + 1, {x, 0, nn}], x] (* Geoffrey Critzer, Nov 02 2014 *)

a(n) = A038057(n)/n = 2^n * n^(n-2) for n>=1. E.g.f. B(2*x) where B(x) is e.g.f. of A000272.

A038061 a(n) = 3^n*n^(n-1).

Original entry on oeis.org

3, 18, 243, 5184, 151875, 5668704, 257298363, 13759414272, 847288609443, 59049000000000, 4594736955793347, 394865111526801408, 37144672966729275363, 3796313155316599873536, 418886580596209716796875

Offset: 1

Christian G. Bower, Jan 04 1999

nonn

Labeled rooted trees with 3-colored nodes.

Index entries for sequences related to rooted trees

Cf. A000169, A038055-A038062.

A038061:=n->3^n*n^(n-1): seq(A038061(n), n=1..20); # Wesley Ivan Hurt, Feb 13 2017

Table[3^n n^(n-1),{n,20}] (* Harvey P. Dale, Aug 24 2012 *)

E.g.f. B(3x) where B(x) is e.g.f. of A000169.

New description from Vladeta Jovovic, Mar 08 2003

A038060 Number of trees with 3-colored nodes.

Original entry on oeis.org

1, 3, 6, 18, 75, 342, 1773, 9894, 58596, 362061, 2314119, 15185223, 101830986, 695253993, 4819762446, 33851833506, 240472935735, 1725315714729, 12487872432924, 91097741283408, 669227023979088, 4947500029023540, 36786363204267282, 274949933519917908

Offset: 0

Christian G. Bower, Jan 04 1999

nonn

Index entries for sequences related to trees

Cf. A000055, A038055-A038062.

Equals 3 * A006965(n).

b[n_] := b[n] = If[n < 2, 3n, (Sum[Sum[b[d] d, {d, Divisors[j]}] b[n - j], {j, 1, n - 1}])/(n - 1)];
a[n_] := If[n == 0, 1, b[n] - (Sum[b[k] b[n - k], {k, 0, n}] - If[Mod[n, 2] == 0, b[n/2], 0])/2];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 01 2020, after Alois P. Heinz in A006965 *)

G.f.: B(x) - B^2(x)/2 + B(x^2)/2, where B(x) is g.f. for A038059.

cp's OEIS Frontend

A038055 Number of n-node rooted trees with nodes of 2 colors.

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A038056 Number of trees with n 2-colored nodes.

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A038057 a(n) = 2^n*n^(n-1).

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A038059 Number of rooted trees with 3-colored nodes.

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A038058 Number of labeled trees with 2-colored nodes.

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A038061 a(n) = 3^n*n^(n-1).

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Maple

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A038060 Number of trees with 3-colored nodes.

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Mathematica

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