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
Examples
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)
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]
Crossrefs
Programs
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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 *) -
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
Extensions
a(53) corrected and terms a(56) and beyond from Andrew Howroyd, Dec 17 2020