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.

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, 60, 11, 23, 142, 404, 656, 656, 404, 142, 23, 46, 334, 1105, 2143, 2652, 2143, 1105, 334, 46, 98, 794, 2995, 6737, 9934, 9934, 6737, 2995, 794, 98, 207, 1888, 7999, 20504, 35080, 41788, 35080, 20504, 7999, 1888, 207

Offset

0,5

Comments

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

First column is A001190.

Row sums are given by A226909.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1274

Example

Triangle begins:
   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,  60,  11;
  23, 142, 404, 656, 656, 404, 142, 23;
  ...

Mathematica

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];
CoefficientList[#, y]& /@ B[11] // Flatten (* Jean-François Alcover, Sep 24 2019, from PARI *)

Prog

(PARI)
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}
{ my(A=B(10)); for(n=1, #A, print(Vec(A[n]))) } \\ Andrew Howroyd, May 21 2018

Crossrefs

Cf. A001190, A226909.

Sequence in context: A137327 A143913 A228815 * A277741 A241138 A241349

Adjacent sequences: A241552 A241553 A241554 * A241556 A241557 A241558

Keyword

nonn,tabl

Author

David Serena, May 17 2014

Extensions

Edited by Nathaniel Johnston, Sep 11 2014

Missing term inserted and a(45) and beyond from Andrew Howroyd, May 21 2018

Status

approved