A144528 Triangle read by rows: T(n,k) is the number of trees on n unlabeled nodes with all nodes of degree <= k (n>=1, 0 <= k <= n-1).

1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 1, 2, 3, 0, 0, 1, 4, 5, 6, 0, 0, 1, 6, 9, 10, 11, 0, 0, 1, 11, 18, 21, 22, 23, 0, 0, 1, 18, 35, 42, 45, 46, 47, 0, 0, 1, 37, 75, 94, 101, 104, 105, 106, 0, 0, 1, 66, 159, 204, 223, 230, 233, 234, 235, 0, 0, 1, 135, 355, 473, 520, 539, 546, 549, 550, 551

Offset

1,10

Links

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

Rebecca Neville, Graphs whose vertices are forests with bounded degree, Graph Theory Notes of New York, LIV (2008), 12-21. [Wayback Machine link]

Example

Triangle begins:
  1
  0 1
  0 0 1
  0 0 1  2
  0 0 1  2  3
  0 0 1  4  5  6
  0 0 1  6  9 10  11
  0 0 1 11 18 21  22  23
  0 0 1 18 35 42  45  46  47
  0 0 1 37 75 94 101 104 105 106
  ...
From Andrew Howroyd, Dec 17 2020: (Start)
Formatted as an array to show the full columns:
================================================
n\k  | 0 1 2   3   4   5   6   7   8   9  10
-----+------------------------------------------
   1 | 1 1 1   1   1   1   1   1   1   1   1 ...
   2 | 0 1 1   1   1   1   1   1   1   1   1 ...
   3 | 0 0 1   1   1   1   1   1   1   1   1 ...
   4 | 0 0 1   2   2   2   2   2   2   2   2 ...
   5 | 0 0 1   2   3   3   3   3   3   3   3 ...
   6 | 0 0 1   4   5   6   6   6   6   6   6 ...
   7 | 0 0 1   6   9  10  11  11  11  11  11 ...
   8 | 0 0 1  11  18  21  22  23  23  23  23 ...
   9 | 0 0 1  18  35  42  45  46  47  47  47 ...
  10 | 0 0 1  37  75  94 101 104 105 106 106 ...
  11 | 0 0 1  66 159 204 223 230 233 234 235 ...
  12 | 0 0 1 135 355 473 520 539 546 549 550 ...
  ...
(End)

Mathematica

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]}]];
b[0, i_, t_, k_] = 1; a = {}; nmax = 20;
For[ni=2, ni < nmax-1, ni++, (* columns 3 to max-1 *)
  gf[x_] = 1 + Sum[b[j-1, j-1, ni, ni] x^j, {j, 1, nmax}];
  ci[x_] = SymmetricGroupIndex[ni+1, x] /. x[i_] -> gf[x^i];
  a = Append[a, CoefficientList[Normal[Series[
    gf[x] - (gf[x]^2 - gf[x^2])/2 + x ci[x], {x, 0, nmax}]], x]]; ]
Join[{1, 0, 1, 0, 0, 1}, Table[Join[{0, 0, 1}, Table[a[[k-3]][[n+1]],
{k, 4, n}]], {n, 4, nmax}]] // Flatten - Robert A. Russell, Feb 05 2023

Prog

(PARI) \\ here V(n, k) gives column k as a vector.
MSet(p, k)={my(n=serprec(p, x)-1); if(min(k, n)<1, 1 + O(x*x^n), polcoef(exp( sum(i=1, min(k, n), (y^i + O(y*y^k))*subst(p + O(x*x^(n\i)), x, x^i)/i ))/(1-y + O(y*y^k)), k, y))}
V(n, k)={my(g=1+O(x)); for(n=2, n, g=x*MSet(g, k-1)); Vec(1 + x*MSet(g, k) + (subst(g, x, x^2) - g^2)/2)}
M(n, m=n)={Mat(vector(m, k, V(n, k-1)[2..1+n]~))}
{ my(T=M(12)); for(n=1, #T~, print(T[n, 1..n])) } \\ Andrew Howroyd, Dec 18 2020

Crossrefs

Columns k=2..7 are A000012, A000672, A000602, A036650, A036653, A359392.

The last three diagonals give A144527, A144520, A000055.

Cf. A144215, A238414, A299038.

Sequence in context: A123679 A167948 A111755 * A290694 A146164 A263141

Adjacent sequences: A144525 A144526 A144527 * A144529 A144530 A144531

Keyword

nonn,tabl

Author

N. J. A. Sloane, Dec 20 2008

Extensions

a(53) corrected and terms a(56) and beyond from Andrew Howroyd, Dec 17 2020

Status

approved