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A241555 Triangle read by rows: Number T(n,k) of 2-colored binary rooted trees with n nodes and exactly k <= n nodes of a specific color.

%I A241555 #23 Mar 07 2025 07:52:06
%S A241555 1,1,1,1,2,1,2,5,5,2,3,11,16,11,3,6,26,50,50,26,6,11,60,143,188,143,
%T A241555 60,11,23,142,404,656,656,404,142,23,46,334,1105,2143,2652,2143,1105,
%U A241555 334,46,98,794,2995,6737,9934,9934,6737,2995,794,98,207,1888,7999,20504,35080,41788,35080,20504,7999,1888,207
%N A241555 Triangle read by rows: Number T(n,k) of 2-colored binary rooted trees with n nodes and exactly k <= n nodes of a specific color.
%C A241555 T(n,k) = T(n,n-k) by definition.
%C A241555 First column is A001190.
%C A241555 Row sums are given by A226909.
%H A241555 Andrew Howroyd, <a href="/A241555/b241555.txt">Table of n, a(n) for n = 0..1274</a>
%H A241555 David Serena and William J. Buchanan, <a href="https://arxiv.org/abs/2503.02663">Equivalence Classes Induced by Binary Tree Isomorphism -- Generating Functions</a>, arXiv:2503.02663 [math.CO], 2025. See p. 17.
%e A241555 Triangle begins:
%e A241555    1;
%e A241555    1,   1;
%e A241555    1,   2,   1;
%e A241555    2,   5,   5,   2;
%e A241555    3,  11,  16,  11,  3;
%e A241555    6,  26,  50,  50,  26,   6;
%e A241555   11,  60, 143, 188, 143,  60,  11;
%e A241555   23, 142, 404, 656, 656, 404, 142, 23;
%e A241555   ...
%t A241555 B[m_] := Module[{u}, u = Table[0, {m}]; u[[1]] = 1; For[n = 1, n <= Length[u] - 1, n++, u[[n + 1]] = (1 + y)*(Sum[u[[i]]*u[[n + 1 - i]], {i, 1, n}] + If[OddQ[n], u[[Quotient[n, 2] + 1]] /. y -> y^2, 0])/2]; u];
%t A241555 CoefficientList[#, y]& /@ B[11] // Flatten (* _Jean-François Alcover_, Sep 24 2019, from PARI *)
%o A241555 (PARI)
%o A241555 B(n)={my(u=vector(n)); u[1]=1; for(n=1, #u-1, u[n+1]=(1+y)*(sum(i=1, n, u[i]*u[n+1-i]) + if(n%2, subst(u[n\2+1], y, y^2)))/2); u}
%o A241555 { my(A=B(10)); for(n=1, #A, print(Vec(A[n]))) } \\ _Andrew Howroyd_, May 21 2018
%Y A241555 Cf. A001190, A226909.
%K A241555 nonn,tabl
%O A241555 0,5
%A A241555 _David Serena_, May 17 2014
%E A241555 Edited by _Nathaniel Johnston_, Sep 11 2014
%E A241555 Missing term inserted and a(45) and beyond from _Andrew Howroyd_, May 21 2018