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

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

Author

R. J. Mathar, Apr 16 2018

Keywords

• nonn
• tabl

Examples

			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,...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275
• Index entries for sequences related to trees

Crossrefs

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

Programs

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

Formula

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

Extensions

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

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