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.

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

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

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

Adjacent sequences: A294780 A294781 A294782 * A294784 A294785 A294786

Keyword

nonn,tabl

Author

R. J. Mathar, Apr 16 2018

Extensions

Row 10 completed. - R. J. Mathar, Apr 29 2018

Status

approved