A124328 Triangle read by rows: T(n,k) is the number of ordered trees with n edges, with thinning limbs and with root of degree k (1<=k<=n). An ordered tree with thinning limbs is such that if a node has k children, all its children have at most k children.

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 10, 9, 4, 1, 1, 22, 25, 14, 5, 1, 1, 46, 69, 44, 20, 6, 1, 1, 101, 186, 137, 70, 27, 7, 1, 1, 220, 503, 416, 235, 104, 35, 8, 1, 1, 492, 1356, 1256, 766, 375, 147, 44, 9, 1, 1, 1104, 3663, 3760, 2465, 1296, 567, 200, 54, 10, 1, 1, 2515, 9907

Offset

1,5

Comments

Row sums yield A124344. T(n,2) = A124329(n).

Links

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

Formula

The g.f. F[k]=F[k](z) of column k satisfies F[k]={F[k-1]^(1/(k-1) + zF[k]}^k; F[1]=z/(1-z).

Central terms are: T(2n-1,n) = A124889(n-1), T(2n,n) = A124891(n-1), for n>=1. - Paul D. Hanna, Nov 12 2006

Example

Triangle starts:
1;
1,1;
1,2,1;
1,5,3,1;
1,10,9,4,1;

Mathematica

t[n_, n_] = 1; t[n_, k_] /; n == k + 1 := t[n, k] = n - 1; t[n_, k_] := t[n, k] = Coefficient[(1 + x*Sum[ x^(r - 1)*Sum[ t[r, c], {c, 1, k }], {r, 1, n - k}] + x^n)^k, x, n - k ]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 21 2013, after Paul D. Hanna *)

Prog

(PARI) {T(n, k)=if(n==k, 1, if(n==k+1, n-1, polcoeff( (1 + x*sum(r=1, n-k, x^(r-1)*sum(c=1, k, T(r, c)))+x*O(x^n))^k, n-k)))} - Paul D. Hanna, Nov 12 2006

Crossrefs

Cf. A124344, A124329.

Cf. A124889, A124890, A124891.

Sequence in context: A137570 A079213 A047884 * A368487 A055818 A106240

Adjacent sequences: A124325 A124326 A124327 * A124329 A124330 A124331

Keyword

nonn,tabl

Author

Emeric Deutsch, Nov 03 2006

Extensions

More terms from Paul D. Hanna, Nov 12 2006

Status

approved