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.
Original entry on oeis.org
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
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,
-
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
A302939
Number of signed trees with n nodes and p positive edges. Triangle T(n,p) read by rows, 0<=p
Original entry on oeis.org
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
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,...
-
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
A331114
Number of rooted trees with 2-colored non-root nodes with an n nodes of each color.
Original entry on oeis.org
1, 3, 37, 596, 11513, 245356, 5597060, 133950215, 3323281496, 84787933926, 2212123329500, 58779046239904, 1585796125188065, 43337540217740908, 1197492197496481744, 33406620056723507124, 939775084670916268134, 26632926871927867655261, 759732892913483065912296
Offset: 0
a(1) = 3 because there are three trees with one root node and one additional node of each color:
1 -- o -- 2,
o -- 1 -- 2,
o -- 2 -- 1.
Showing 1-3 of 3 results.
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