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A336524 Triangular array read by rows. T(n,k) is the number of unlabeled binary trees with n internal nodes and exactly k distinguished external nodes (leaves) for 0 <= k <= n+1 and n >= 0.

%I A336524 #23 Oct 19 2020 11:26:46
%S A336524 1,1,1,2,1,2,6,6,2,5,20,30,20,5,14,70,140,140,70,14,42,252,630,840,
%T A336524 630,252,42,132,924,2772,4620,4620,2772,924,132,429,3432,12012,24024,
%U A336524 30030,24024,12012,3432,429
%N A336524 Triangular array read by rows. T(n,k) is the number of unlabeled binary trees with n internal nodes and exactly k distinguished external nodes (leaves) for 0 <= k <= n+1 and n >= 0.
%H A336524 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 213.
%F A336524 O.g.f. for column k: 1/k!*(d/dy)^k y*B(y*x)|y=1 where B(x) is the o.g.f. for A000108.
%F A336524 From _Vladimir Kruchinin_, Oct 16 2020: (Start)
%F A336524 O.g.f.: (1-sqrt(-4*x*y-4*x+1))/(2*x).
%F A336524 T(n,m) = C(n+m,n)*C(2*n+1,n+m)/(2*n+1).
%F A336524 (End)
%e A336524 Taylor series starts: (y + 1) + x*(y + 1)^2 + 2*x^2*(y + 1)^3 + 5*x^3*(y + 1)^4 + 14*x^4*(y + 1)^5 + ...
%e A336524 Triangle T(n, k) begins:
%e A336524    1,   1;
%e A336524    1,   2,   1;
%e A336524    2,   6,   6,   2;
%e A336524    5,  20,  30,  20,   5;
%e A336524   14,  70, 140, 140,  70,  14;
%e A336524   42, 252, 630, 840, 630, 252, 42;
%e A336524   ...
%t A336524 nn = 5; b[z_] := (1 - Sqrt[1 - 4 z])/(2 z);Map[Select[#, # > 0 &] &,Transpose[Table[CoefficientList[Series[D[v b[v z], {v, k}]/k! /. v -> 1, {z, 0, nn}], z], {k, 0, nn + 1}]]] // Grid
%o A336524 (Maxima)
%o A336524 T(n,m):=(binomial(n+m,n)*binomial(2*n+1,n+m))/(2*n+1); /* _Vladimir Kruchinin_, Oct 16 2020 */
%o A336524 (PARI) for(n=1,8,for(k=0,n,print1(binomial(n,k)*binomial(2*n-2,n-1)/n,", "));print()) \\ _Hugo Pfoertner_, Oct 16 2020
%Y A336524 Cf. A025225 (row sums), A000108 (column k=0), A000984 (column k=1), A002457 (column k=2).
%Y A336524 Cf. A007318.
%K A336524 nonn,tabf
%O A336524 0,4
%A A336524 _Geoffrey Critzer_, Jul 24 2020