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A144528
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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).
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11
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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
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OFFSET
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1,10
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LINKS
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EXAMPLE
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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
...
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)
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MATHEMATICA
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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]],
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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a(53) corrected and terms a(56) and beyond from Andrew Howroyd, Dec 17 2020
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STATUS
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approved
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