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A088326
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Triangle T(n,k) (n>=1, 1<=k<=n) read by rows, giving number of Piet Hut's "coat-hanger" arrangements: unlabeled forests of rooted trees with n edges and k connected components, in which the outdegree of each node is <= 2 and the symmetric group acts on the components.
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3
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1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 6, 5, 3, 1, 1, 11, 12, 6, 3, 1, 1, 23, 23, 14, 6, 3, 1, 1, 46, 52, 29, 15, 6, 3, 1, 1, 98, 109, 68, 31, 15, 6, 3, 1, 1, 207, 244, 147, 74, 32, 15, 6, 3, 1, 1, 451, 532, 337, 163, 76, 32, 15, 6, 3, 1, 1, 983, 1196, 757, 380, 169, 77, 32, 15, 6, 3, 1, 1
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OFFSET
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1,4
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LINKS
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FORMULA
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G.f.: exp(sum_{k=1..infinity) z^k*B(x^k)/k ), where B(x) = x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 11*x^6 + ... = G001190(x)/x - 1 and G001190 is the g.f. for the Wedderburn-Etherington numbers A001190.
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EXAMPLE
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Triangle begins
1
1 1
2 1 1
3 3 1 1
6 5 3 1 1
11 12 6 3 1 1
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MAPLE
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g:= proc(n) option remember; `if`(n<2, n, `if`(n::odd, 0,
(t-> t*(1-t)/2)(g(n/2)))+add(g(i)*g(n-i), i=1..n/2))
end:
b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
`if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*binomial(
g(i+1)+j-1, j), j=0..min(n/i, p)))))
end:
T:= (n, k)-> b(n$2, k):
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MATHEMATICA
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g[n_] := g[n] = If[n<2, n, If[OddQ[n], 0, Function[t, t*(1-t)/2][g[n/2]]] + Sum[g[i]*g[n - i], {i, 1, n/2}]];
b[n_, i_, p_] := b[n, i, p] = If[p>n, 0, If[n == 0, 1, If[Min[i, p]<1, 0, Sum[b[n-i*j, i-1, p-j]*Binomial[g[i+1]+j-1, j], {j, 0, Min[n/i, p]}]]]];
T[n_, k_] := b[n, n, k];
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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