A294783 - OEIS

%I #67 May 14 2018 16:59:39

%S 1,1,1,1,1,1,1,2,2,1,2,4,6,4,2,3,9,15,15,9,3,6,20,43,51,43,20,6,11,48,

%T 116,175,175,116,48,11,23,115,329,573,698,573,329,115,23,47,286,918,

%U 1866,2626,2626,1866,918,286,47,106,719,2609,5978,9656,11241,9656,5978,2609,719,106,235,1842

%N Number of trees with n bicolored nodes and f nodes of the first color. Triangle T(n,f) read by rows, 0<=f<=n.

%H Andrew Howroyd, <a href="/A294783/b294783.txt">Table of n, a(n) for n = 0..1274</a>

%F T(n,f) = T(n,n-f), flipping all node colors.

%e The triangle starts

%e     1;

%e     1,   1;

%e     1,   1,   1;

%e     1,   2,   2,    1;

%e     2,   4,   6,    4,    2;

%e     3,   9,  15,   15,    9,    3;

%e     6,  20,  43,   51,   43,   20,    6;

%e    11,  48, 116,  175,  175,  116,   48,  11;

%e    23, 115, 329,  573,  698,  573,  329, 115,  23;

%e    47, 286, 918, 1866, 2626, 2626, 1866, 918, 286, 47;

%e   106, 719,2609, 5978, 9656,11241, 9656,5978,2609,719,106;

%e   235,1842,

%o (PARI)

%o R(n, y)={my(v=vector(n)); v[1]=1; for(k=1, n-1, my(p=(1+y)*v[k]); my(q=Vec(prod(j=0, poldegree(p,y), (1/(1-x*y^j) + O(x*x^(n\k)))^polcoeff(p,j)))); v=vector(n, j, v[j] + sum(i=1, (j-1)\k, v[j-i*k] * q[i+1]))); v;}

%o M(n)={my(B=(1+y)*x*Ser(R(n,y))); 1 + B - (B^2 - substvec(B, [x,y], [x^2,y^2]))/2}

%o { my(A=M(10)); for(n=0, #A-1, print(Vecrev(polcoeff(A, n)))) } \\ _Andrew Howroyd_, May 12 2018

%Y Cf. A038056 (row sums), A000055 (diagonal and 1st column), A000081 (subdiagonal and 2nd column), A303833 (3rd column), A303843 (4th column), A304311 (connected graphs), A304489 (rooted).

%K nonn,tabl

%O 0,8

%A _R. J. Mathar_, Apr 16 2018

%E Row 10 completed. - _R. J. Mathar_, Apr 29 2018