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A088326 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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

FORMULA

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.

G.f.: Product_{j>=1} 1/(1-y*x^j)^A001190(j+1). - Alois P. Heinz, Sep 11 2017

EXAMPLE

See A088325 for illustration.

Triangle begins

1

1 1

2 1 1

3 3 1 1

6 5 3 1 1

11 12 6 3 1 1

MAPLE

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):

seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Sep 11 2017

MATHEMATICA

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];

Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-Fran├žois Alcover, Jun 11 2018, after Alois P. Heinz *)

CROSSREFS

First 3 columns are A001190, A036657, A036658.

Row sums are A088325.

T(2n,n) gives A305839.

Sequence in context: A090011 A061554 A296373 * A216956 A285522 A181039

Adjacent sequences:  A088323 A088324 A088325 * A088327 A088328 A088329

KEYWORD

nonn,tabl,easy

AUTHOR

N. J. A. Sloane, Nov 06 2003

EXTENSIONS

More terms from Vladeta Jovovic, Nov 06 2003

STATUS

approved

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Last modified February 18 00:19 EST 2019. Contains 320237 sequences. (Running on oeis4.)