A255523 -id:A255523 - cp's OEIS frontend

A255517 Number A(n,k) of rooted identity trees with n nodes and k-colored non-root nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 3, 5, 2, 0, 0, 1, 4, 12, 18, 3, 0, 0, 1, 5, 22, 64, 66, 6, 0, 0, 1, 6, 35, 156, 363, 266, 12, 0, 0, 1, 7, 51, 310, 1193, 2214, 1111, 25, 0, 0, 1, 8, 70, 542, 2980, 9748, 14043, 4792, 52, 0, 0, 1, 9, 92, 868, 6273, 30526, 82916, 91857, 21124, 113, 0

Offset: 0

Alois P. Heinz, Feb 24 2015

From Vaclav Kotesovec, Feb 24 2015: (Start)
k    Limit n->infinity A(n,k)^(1/n)
  2.517540352632003890795354598463447277335981266803... = A246169
  5.249032491228170579164952216184309265343086337648... = A246312
  7.969494030514425004826375511986491746399264355846...
 10.688492754969652458452048798468242930479212456958...
 13.407087472537747579787047072702638639945914705837...
 16.125529360448558670505097146631763969697822205298...
 18.843901825822305757579605844910623225182677164912...
 21.562238702430237066018783115405680041128676137631...
 24.280555694806692616578932533497629224907619468796...
26.998860838916733933849490675388336975888308433826...
271.64425688361559470587959030374804709717287744789...
Conjecture: For big k the limit asymptotically approaches k*exp(1).
(End)

			A(3,2) = 5:
  o    o    o    o      o
  |    |    |    |     / \
  1    1    2    2    1   2
  |    |    |    |
  1    2    1    2
Square array A(n,k) begins:
  0,  0,   0,    0,    0,     0,     0, ...
  1,  1,   1,    1,    1,     1,     1, ...
  0,  1,   2,    3,    4,     5,     6, ...
  0,  1,   5,   12,   22,    35,    51, ...
  0,  2,  18,   64,  156,   310,   542, ...
  0,  3,  66,  363, 1193,  2980,  6273, ...
  0,  6, 266, 2214, 9748, 30526, 77262, ...

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

Columns k=1-10 give: A004111, A005753, A052757, A052772, A052797, A255518, A255519, A255520, A255521, A255522.
Rows n=0-4 give: A000004, A000012, A001477, A000326, 2*A051662(k-1) for k>0.
Lower diagonal gives A255523.
Cf. A242249, A256068.

with(numtheory):
A:= proc(n, k) option remember; `if`(n<2, n, add(A(n-j, k)*add(
      k*A(d, k)*d*(-1)^(j/d+1), d=divisors(j)), j=1..n-1)/(n-1))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);

A[n_, k_] := A[n, k] = If[n<2, n, Sum[A[n-j, k]*Sum[k*A[d, k]*d*(-1)^(j/d + 1), {d, Divisors[j]}], {j, 1, n-1}]/(n-1)]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)

A242375 Number of rooted trees with n n-colored non-root nodes.

Original entry on oeis.org

1, 1, 7, 82, 1499, 37476, 1200705, 46990952, 2175619923, 116400215521, 7069820334023, 480722969498938, 36186340018129392, 2987845924408179654, 268530017303221572650, 26098422892000807053155, 2727654868575748827350403, 305075571192329680642519141

Offset: 0

Alois P. Heinz, May 12 2014

nonn

			a(2) = 7:
  o    o    o    o      o        o        o
  |    |    |    |     / \      / \      / \
  1    1    2    2    1   1    1   2    2   2
  |    |    |    |
  1    2    1    2

Alois P. Heinz, Table of n, a(n) for n = 0..200

A diagonal of A242249.

Cf. A255523.

with(numtheory):
b:= proc(n, k) option remember; `if`(n<2, n, (add(add(d*
      b(d, k), d=divisors(j))*b(n-j, k)*k, j=1..n-1))/(n-1))
    end:
a:= n-> b(n+1, n):
seq(a(n), n=0..20);

b[n_, k_] := b[n, k] = If[n < 2, n, (Sum[Sum[d*b[d, k], {d, Divisors[j]}] * b[n - j, k]*k, {j, 1, n - 1}])/(n - 1)];
a[n_] := b[n + 1, n];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 21 2017, translated from Maple *)

a(n) ~ c * exp(n) * n^(n-3/2), where c = exp(1 + exp(-2)/2) / sqrt(2*Pi) = 1.160358615244339554387715748... . - Vaclav Kotesovec, Aug 28 2014, updated Mar 18 2024

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A255517 Number A(n,k) of rooted identity trees with n nodes and k-colored non-root nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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