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A304489
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Triangle read by rows: T(n,k) = number of rooted signed trees with n nodes and k positive edges (0 <= k < n).
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4
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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
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OFFSET
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1,4
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COMMENTS
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Equivalently, the number of rooted trees with 2-colored non-root nodes, n nodes and k nodes of the first color.
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LINKS
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EXAMPLE
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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;
...
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PROG
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(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]))) }
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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