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
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