A329054
Array read by antidiagonals: T(n,m) is the number of unlabeled bicolored trees with n nodes of one color and m of the other.
Original entry on oeis.org
1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 4, 2, 1, 0, 0, 1, 3, 7, 7, 3, 1, 0, 0, 1, 3, 10, 14, 10, 3, 1, 0, 0, 1, 4, 14, 28, 28, 14, 4, 1, 0, 0, 1, 4, 19, 45, 65, 45, 19, 4, 1, 0, 0, 1, 5, 24, 73, 132, 132, 73, 24, 5, 1, 0
Offset: 0
Array begins:
===================================================
n\m | 0 1 2 3 4 5 6 7 8
----+----------------------------------------------
0 | 1, 1, 0, 0, 0, 0, 0, 0, 0, ...
1 | 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2 | 0, 1, 1, 2, 2, 3, 3, 4, 4, ...
3 | 0, 1, 2, 4, 7, 10, 14, 19, 24, ...
4 | 0, 1, 2, 7, 14, 28, 45, 73, 105, ...
5 | 0, 1, 3, 10, 28, 65, 132, 242, 412, ...
6 | 0, 1, 3, 14, 45, 132, 316, 693, 1349, ...
7 | 0, 1, 4, 19, 73, 242, 693, 1742, 3927, ...
8 | 0, 1, 4, 24, 105, 412, 1349, 3927, 10079, ...
...
The equivalent array for labeled nodes is
A072590.
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EulerXY(A)={my(j=serprec(A,x)); exp(sum(i=1, j, 1/i * subst(subst(A + x * O(x^(j\i)), x, x^i), y, y^i)))}
R(n)={my(A=O(x)); for(j=1, 2*n, A = if(j%2, 1, y)*x*EulerXY(A)); A};
P(n)={my(r1=R(n), r2=x*EulerXY(r1), s=r1+r2-r1*r2); Vec(1 + s)}
{ my(A=P(10)); for(n=0, #A\2, for(k=0, #A\2, print1(polcoef(A[n+k+1], k), ", ")); print) }
A119856
Number of equicolorable (unrooted) trees on 2n nodes.
Original entry on oeis.org
1, 1, 3, 9, 37, 168, 895, 5097, 30983, 196096, 1283552, 8621364, 59176966, 413613891, 2936303012, 21128390679, 153841228779, 1131941424480, 8406680733066, 62958726953945, 475074493277317, 3609405128045162, 27593870865196624, 212159118489924538, 1639760091688265416
Offset: 1
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\\ R is b.g.f of rooted trees x nodes, y in one part
R(n)={my(A=O(x)); for(j=1, 2*n, A = if(j%2,1,y)*x*exp(sum(i=1, j, 1/i * subst(subst(A + x * O(x^(j\i)), x, x^i), y, y^i)))); A};
seq(n)={my(A=Pol(R(n))); my(r(x,y)=substvec(A, ['x,'y], [x,y/x])); Vec(polcoeff((r(x,y/x) + r(y/x,x) - r(x,y/x)*r(y/x,x)), 0) + O(y*y^n) + r(y,y))/2} \\ Andrew Howroyd, May 23 2018
A119855
Number of equicolorable rooted trees on 2n nodes.
Original entry on oeis.org
1, 2, 9, 44, 249, 1506, 9687, 64803, 447666, 3169566, 22897260, 168168164, 1252391041, 9437809359, 71850420813, 551876468717, 4272100488830, 33299732401378, 261165251593743, 2059638535690473, 16324255856903830, 129969379170062142, 1039056925387672998
Offset: 1
- N. Pippenger, Enumeration of equicolorable trees, SIAM J. Discrete Math., 14 (2001), 93-115.
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\\ R is b.g.f of rooted trees x nodes, y in one part
R(n)={my(A=O(x)); for(j=1, 2*n, A = if(j%2,1,y)*x*exp(sum(i=1, j, 1/i * subst(subst(A + x * O(x^(j\i)), x, x^i), y, y^i)))); A};
seq(n)={my(A=Pol(R(n))); my(r(x,y)=substvec(A, ['x,'y], [x,y/x])); Vec(polcoeff(r(x, y/x), 0) + O(y*y^n))} \\ Andrew Howroyd, May 23 2018
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