A302939 Number of signed trees with n nodes and p positive edges. Triangle T(n,p) read by rows, 0<=p<n.

1, 1, 1, 1, 1, 1, 2, 3, 3, 2, 3, 6, 9, 6, 3, 6, 16, 27, 27, 16, 6, 11, 37, 79, 96, 79, 37, 11, 23, 96, 233, 349, 349, 233, 96, 23, 47, 239, 679, 1187, 1439, 1187, 679, 239, 47, 106, 622, 1987, 4017, 5639, 5639, 4017, 1987, 622, 106, 235, 1607, 5784, 13216, 21263, 24758, 21263, 13216, 5784, 1607, 235

Offset

1,7

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Index entries for sequences related to trees

Formula

T(n,p) = T(n,n-p-1), flipping all edge signs.

Example

T(2,0)=T(2,1)=1: the tree on 2 nodes (one edge) has one variant with no positive edge and one variant with one positive edge.
T(4,1)=3: the 2 trees on 4 nodes (three edges) have two variants from the linear tree with a positive edge (edge in the middle or at the end) and one variant from the star graph with one positive edge.
T(5,0)=3: there are 3 trees on 5 nodes (4 edges) where all edges are negative.
The triangle starts
    1;
    1,   1;
    1,   1,   1;
    2,   3,   3,    2;
    3,   6,   9,    6,    3;
    6,  16,  27,   27,   16,    6;
   11,  37,  79,   96,   79,   37,  11;
   23,  96, 233,  349,  349,  233,  96,  23;
   47, 239, 679, 1187, 1439, 1187, 679, 239, 47;
  106, 622,...

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=x*Ser(R(n, y))); B - (1+y)*(B^2 - substvec(B, [x, y], [x^2, y^2]))/2}
{ my(A=Vec(M(10))); for(n=1, #A, print(Vecrev(A[n]))) } \\ Andrew Howroyd, May 13 2018

Crossrefs

Cf. A000060 (row sums), A000055 (diagonal and 1st column), A027852 (subdiagonal and 2nd column), A304489 (rooted), A331113 (central coefficients).

Sequence in context: A105612 A141744 A089783 * A090414 A068227 A235343

Adjacent sequences: A302936 A302937 A302938 * A302940 A302941 A302942

Keyword

nonn,tabl

Author

R. J. Mathar, Apr 16 2018

Extensions

Completed row 10. - R. J. Mathar, Apr 29 2018

Terms a(58) and beyond from Andrew Howroyd, May 13 2018

Status

approved