cp's OEIS Frontend

A361357 Triangle read by rows: T(n,k) is the number of noncrossing caterpillars with n edges and diameter k, 0 <= k <= n.

%I A361357 #19 Mar 25 2025 08:55:27
%S A361357 1,0,1,0,0,3,0,0,4,8,0,0,5,30,20,0,0,6,75,144,48,0,0,7,154,595,504,
%T A361357 112,0,0,8,280,1848,2896,1536,256,0,0,9,468,4788,12060,11268,4320,576,
%U A361357 0,0,10,735,10920,40700,58760,38480,11520,1280
%N A361357 Triangle read by rows: T(n,k) is the number of noncrossing caterpillars with n edges and diameter k, 0 <= k <= n.
%C A361357 A noncrossing caterpillar is a noncrossing tree that is a caterpillar tree (also called a caterpillar graph).
%C A361357 The diameter of a tree is also the length of the longest path.
%H A361357 Andrew Howroyd, <a href="/A361357/b361357.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H A361357 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CaterpillarGraph.html">Caterpillar Graph</a>.
%H A361357 Wikipedia, <a href="https://en.wikipedia.org/wiki/Caterpillar_tree">Caterpillar tree</a>.
%F A361357 T(n,2) = n + 1 for n >= 2.
%F A361357 G.f.: A(x,y) = 1 + x*y + (x*y)^2*((3 - 2*x) + (4 - 3*x + x^2)*F(x,y) + (1 + 2*x)*F(x,y)^2)/(1 - x)^2 where F(x,y) = x*y*(2 - x)/(1 - (3 + 2*y)*x + 3*x^2 - x^3).
%e A361357 Triangle begins:
%e A361357   1;
%e A361357   0, 1;
%e A361357   0, 0,  3;
%e A361357   0, 0,  4,   8;
%e A361357   0, 0,  5,  30,    20;
%e A361357   0, 0,  6,  75,   144,    48;
%e A361357   0, 0,  7, 154,   595,   504,   112;
%e A361357   0, 0,  8, 280,  1848,  2896,  1536,   256;
%e A361357   0, 0,  9, 468,  4788, 12060, 11268,  4320,   576;
%e A361357   0, 0, 10, 735, 10920, 40700, 58760, 38480, 11520, 1280;
%e A361357   ...
%o A361357 (PARI)
%o A361357 T(n) = {my(f=x*y*(2 - x)/(1 - (3 + 2*y)*x + 3*x^2 - x^3), g = 1 + x*y + (x*y)^2*((3 - 2*x) + (4 - 3*x + x^2)*f + (1 + 2*x)*f^2)/(1 - x)^2); [Vecrev(p) | p<-Vec(g + O(x*x^n))]}
%o A361357 { my(A=T(9)); for(i=1, #A, print(A[i])) }
%Y A361357 Row sums are A361356.
%Y A361357 Main diagonal is A001792(n-1).
%K A361357 nonn,tabl
%O A361357 0,6
%A A361357 _Andrew Howroyd_, Mar 09 2023