A088326 - OEIS

%I #16 Nov 27 2024 12:08:40

%S 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,

%T 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,

%U 451,532,337,163,76,32,15,6,3,1,1,983,1196,757,380,169,77,32,15,6,3,1,1

%N 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.

%H Alois P. Heinz, <a href="/A088326/b088326.txt">Rows n = 1..141, flattened</a>

%F 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.

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

%e See A088325 for illustration.

%e Triangle begins

%e    1

%e    1  1

%e    2  1 1

%e    3  3 1 1

%e    6  5 3 1 1

%e   11 12 6 3 1 1

%e   ...

%p g:= proc(n) option remember; `if`(n<2, n, `if`(n::odd, 0,

%p       (t-> t*(1-t)/2)(g(n/2)))+add(g(i)*g(n-i), i=1..n/2))

%p     end:

%p b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,

%p       `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*binomial(

%p        g(i+1)+j-1, j), j=0..min(n/i, p)))))

%p     end:

%p T:= (n, k)-> b(n$2, k):

%p seq(seq(T(n, k), k=1..n), n=1..14);  # _Alois P. Heinz_, Sep 11 2017

%t 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}]];

%t 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 T[n_, k_] := b[n, n, k];

%t Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 11 2018, after _Alois P. Heinz_ *)

%Y First 3 columns are A001190, A036657, A036658.

%Y Row sums are A088325.

%Y T(2n,n) gives A305839.

%K nonn,tabl,easy,changed

%O 1,4

%A _N. J. A. Sloane_, Nov 06 2003

%E More terms from _Vladeta Jovovic_, Nov 06 2003