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
Examples
Triangle starts: 1; 1,1; 1,2,1; 1,5,3,1; 1,10,9,4,1;
Links
- Alois P. Heinz, Rows n = 1..141, flattened
Programs
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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 *) -
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
Formula
The g.f. F[k]=F[k](z) of column k satisfies F[k]=(F[k-1]^(1/(k-1)) + z*F[k])^k; F[1]=z/(1-z).
Central terms are: T(2*n-1,n) = A124889(n-1), T(2*n,n) = A124891(n-1), for n>=1. - Paul D. Hanna, Nov 12 2006
Extensions
More terms from Paul D. Hanna, Nov 12 2006
Comments