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

%I A304489 #13 Jan 10 2020 16:21:11
%S A304489 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,
%T A304489 469,214,48,115,612,1577,2445,2445,1577,612,115,286,1747,5204,9480,
%U A304489 11513,9480,5204,1747,286,719,4995,16865,35357,50363,50363,35357,16865,4995,719
%N A304489 Triangle read by rows: T(n,k) = number of rooted signed trees with n nodes and k positive edges (0 <= k < n).
%C A304489 Equivalently, the number of rooted trees with 2-colored non-root nodes, n nodes and k nodes of the first color.
%H A304489 Andrew Howroyd, <a href="/A304489/b304489.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)
%e A304489 Triangle begins:
%e A304489     1;
%e A304489     1,    1;
%e A304489     2,    3,    2;
%e A304489     4,    9,    9,    4;
%e A304489     9,   26,   37,   26,     9;
%e A304489    20,   75,  134,  134,    75,   20;
%e A304489    48,  214,  469,  596,   469,  214,   48;
%e A304489   115,  612, 1577, 2445,  2445, 1577,  612,  115;
%e A304489   286, 1747, 5204, 9480, 11513, 9480, 5204, 1747, 286;
%e A304489   ...
%o A304489 (PARI)
%o A304489 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; }
%o A304489 { my(A=R(10,y)); for(n=1, #A, print(Vecrev(A[n]))) }
%o A304489 (PARI)
%o A304489 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)}
%o A304489 R(n, y)={my(v=[1]); for(k=2,n,v=concat([1], EulerMT(v*(1+y)))); v}
%o A304489 { my(A=R(10,y)); for(n=1, #A, print(Vecrev(A[n]))) }
%Y A304489 Row sums are A000151.
%Y A304489 Columns k=0..1 are A000081, A000243.
%Y A304489 Cf. A294783, A302939, A331114.
%K A304489 nonn,tabl
%O A304489 1,4
%A A304489 _Andrew Howroyd_, May 13 2018