A304489 Triangle read by rows: T(n,k) = number of rooted signed trees with n nodes and k positive edges (0 <= k < n).

1, 1, 1, 2, 3, 2, 4, 9, 9, 4, 9, 26, 37, 26, 9, 20, 75, 134, 134, 75, 20, 48, 214, 469, 596, 469, 214, 48, 115, 612, 1577, 2445, 2445, 1577, 612, 115, 286, 1747, 5204, 9480, 11513, 9480, 5204, 1747, 286, 719, 4995, 16865, 35357, 50363, 50363, 35357, 16865, 4995, 719

Offset: 1

Author

Andrew Howroyd, May 13 2018

Keywords

• nonn
• tabl

Comments

Equivalently, the number of rooted trees with 2-colored non-root nodes, n nodes and k nodes of the first color.

Examples

			Triangle begins:
    1;
    1,    1;
    2,    3,    2;
    4,    9,    9,    4;
    9,   26,   37,   26,     9;
   20,   75,  134,  134,    75,   20;
   48,  214,  469,  596,   469,  214,   48;
  115,  612, 1577, 2445,  2445, 1577,  612,  115;
  286, 1747, 5204, 9480, 11513, 9480, 5204, 1747, 286;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Crossrefs

Row sums are A000151.

Columns k=0..1 are A000081, A000243.

Cf. A294783, A302939, A331114.

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; }
{ my(A=R(10,y)); for(n=1, #A, print(Vecrev(A[n]))) }

PARI

EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
R(n, y)={my(v=[1]); for(k=2,n,v=concat([1], EulerMT(v*(1+y)))); v}
{ my(A=R(10,y)); for(n=1, #A, print(Vecrev(A[n]))) }