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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.

%I A124328 #21 Apr 07 2025 23:39:23
%S A124328 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,
%T A124328 101,186,137,70,27,7,1,1,220,503,416,235,104,35,8,1,1,492,1356,1256,
%U A124328 766,375,147,44,9,1,1,1104,3663,3760,2465,1296,567,200,54,10,1,1,2515,9907
%N 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.
%C A124328 Row sums yield A124344. T(n,2) = A124329(n).
%H A124328 Alois P. Heinz, <a href="/A124328/b124328.txt">Rows n = 1..141, flattened</a>
%F A124328 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).
%F A124328 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
%e A124328 Triangle starts:
%e A124328   1;
%e A124328   1,1;
%e A124328   1,2,1;
%e A124328   1,5,3,1;
%e A124328   1,10,9,4,1;
%t A124328 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_ *)
%o A124328 (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
%Y A124328 Cf. A124344, A124329.
%Y A124328 Cf. A124889, A124890, A124891.
%K A124328 nonn,tabl
%O A124328 1,5
%A A124328 _Emeric Deutsch_, Nov 03 2006
%E A124328 More terms from _Paul D. Hanna_, Nov 12 2006