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A294783
Number of trees with n bicolored nodes and f nodes of the first color. Triangle T(n,f) read by rows, 0<=f<=n.
7
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, 116, 175, 175, 116, 48, 11, 23, 115, 329, 573, 698, 573, 329, 115, 23, 47, 286, 918, 1866, 2626, 2626, 1866, 918, 286, 47, 106, 719, 2609, 5978, 9656, 11241, 9656, 5978, 2609, 719, 106, 235, 1842
OFFSET
0,8
LINKS
FORMULA
T(n,f) = T(n,n-f), flipping all node colors.
EXAMPLE
The triangle starts
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, 116, 175, 175, 116, 48, 11;
23, 115, 329, 573, 698, 573, 329, 115, 23;
47, 286, 918, 1866, 2626, 2626, 1866, 918, 286, 47;
106, 719,2609, 5978, 9656,11241, 9656,5978,2609,719,106;
235,1842,
PROG
(PARI)
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; }
M(n)={my(B=(1+y)*x*Ser(R(n, y))); 1 + B - (B^2 - substvec(B, [x, y], [x^2, y^2]))/2}
{ my(A=M(10)); for(n=0, #A-1, print(Vecrev(polcoeff(A, n)))) } \\ Andrew Howroyd, May 12 2018
CROSSREFS
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).
Sequence in context: A294600 A247495 A230290 * A172021 A325182 A215959
KEYWORD
nonn,tabl
AUTHOR
R. J. Mathar, Apr 16 2018
EXTENSIONS
Row 10 completed. - R. J. Mathar, Apr 29 2018
STATUS
approved