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A204387 Triangle read by rows: T(n,k) is number of noncrossing trees with k edges and path-length n, n >= 1, 1 <= k <= n.

%I A204387 #20 Nov 19 2024 22:12:49
%S A204387 1,0,1,0,2,1,0,0,4,1,0,0,3,6,1,0,0,4,10,8,1,0,0,0,12,21,10,1,0,0,0,12,
%T A204387 32,36,12,1,0,0,0,6,45,72,55,14,1,0,0,0,8,36,119,140,78,16,1,0,0,0,0,
%U A204387 46,144,270,244,105,18,1,0,0,0,0,32,164,416,550,392,136,20,1
%N A204387 Triangle read by rows: T(n,k) is number of noncrossing trees with k edges and path-length n, n >= 1, 1 <= k <= n.
%C A204387 The number of nodes is k + 1. The path-length is the sum of the distances of all nodes from the root node. - _Andrew Howroyd_, Nov 19 2024
%H A204387 Andrew Howroyd, <a href="/A204387/b204387.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%H A204387 E. Deutsch and M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(01)00366-1">Statistics on non-crossing trees</a>, Discrete Math., 254 (2002), 75-87 (see Th. 6).
%F A204387 From _Andrew Howroyd_, Nov 19 2024: (Start)
%F A204387 G.f.: A(x,y) satisfies A(x,y) = 1/(1 - x*y*A(x,x*y)^2).
%F A204387 T(k*(k+1)/2, k) = 2^(k-1).
%F A204387 T(n,k) = 0 for n > k*(k+1)/2.
%F A204387 Sum_{n>=1} n*T(n,k) = A062236(k). (End)
%e A204387 Triangle begins:
%e A204387 1
%e A204387 0 1
%e A204387 0 2 1
%e A204387 0 0 4 1
%e A204387 0 0 3 6 1
%e A204387 0 0 4 10 8 1
%e A204387 0 0 0 12 21 10 1
%e A204387 0 0 0 12 32 36 12 1
%o A204387 (PARI) T(n)={my(g=1+O(x)); for(i=1, n, g=1/(1 - x*y*subst(g,y,x*y)^2)); [Vecrev(p/y) | p<-Vec(g-1)]}
%o A204387 {my(A=T(10)); for(i=1, #A, print(A[i]))} \\ _Andrew Howroyd_, Nov 19 2024
%Y A204387 Row sums are A132332.
%Y A204387 Column sums are A001764.
%Y A204387 Cf. A062236.
%K A204387 nonn,tabl
%O A204387 1,5
%A A204387 _N. J. A. Sloane_, Jan 17 2012
%E A204387 a(34) corrected and a(42) onwards from _Andrew Howroyd_, Nov 19 2024