A241555 - OEIS

%I #18 Sep 24 2019 05:04:07

%S 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 60,11,23,142,404,656,656,404,142,23,46,334,1105,2143,2652,2143,1105,

%U 334,46,98,794,2995,6737,9934,9934,6737,2995,794,98,207,1888,7999,20504,35080,41788,35080,20504,7999,1888,207

%N 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 T(n,k) = T(n,n-k) by definition.

%C First column is A001190.

%C Row sums are given by A226909.

%H Andrew Howroyd, <a href="/A241555/b241555.txt">Table of n, a(n) for n = 0..1274</a>

%e Triangle begins:

%e    1;

%e    1,   1;

%e    1,   2,   1;

%e    2,   5,   5,   2;

%e    3,  11,  16,  11,  3;

%e    6,  26,  50,  50,  26,   6;

%e   11,  60, 143, 188, 143,  60,  11;

%e   23, 142, 404, 656, 656, 404, 142, 23;

%e   ...

%t 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 CoefficientList[#, y]& /@ B[11] // Flatten (* _Jean-François Alcover_, Sep 24 2019, from PARI *)

%o (PARI)

%o 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 { my(A=B(10)); for(n=1, #A, print(Vec(A[n]))) } \\ _Andrew Howroyd_, May 21 2018

%Y Cf. A001190, A226909.

%K nonn,tabl

%O 0,5

%A _David Serena_, May 17 2014

%E Edited by _Nathaniel Johnston_, Sep 11 2014

%E Missing term inserted and a(45) and beyond from _Andrew Howroyd_, May 21 2018