A339830
Number of bicolored trees on n unlabeled nodes such that black nodes are not adjacent to each other.
Original entry on oeis.org
1, 2, 2, 4, 10, 26, 75, 234, 768, 2647, 9466, 34818, 131149, 503640, 1965552, 7777081, 31138051, 125961762, 514189976, 2115922969, 8769932062, 36584593158, 153510347137, 647564907923, 2744951303121, 11687358605310, 49965976656637, 214423520420723, 923399052307921
Offset: 0
a(2) = 2 because at most one node can be colored black.
a(3) = 4 because the only tree is the path graph P_3. If the center node is colored black then neither of the ends can be colored black; otherwise zero, one or both of the ends can be colored black. In total there are 4 possibilities.
There are 3 trees with 5 nodes:
o o
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o---o---o o---o---o---o---o o---o---o
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o o
These correspond respectively to 11, 9 and 6 bicolored trees (with black nodes not adjacent), so a(5) = 11 + 9 + 6 = 26.
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EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(u=v=[1]); for(n=2, n, my(t=concat([1], EulerT(v))); v=concat([1], EulerT(u+v)); u=t); my(g=x*Ser(u+v), gu=x*Ser(u)); Vec(1 + g + (subst(g,x,x^2) - subst(gu,x,x^2) - g^2 + gu^2)/2)}
A339834
Number of bicolored trees on n unlabeled nodes such that every white node is adjacent to a black node.
Original entry on oeis.org
1, 1, 2, 4, 11, 29, 91, 299, 1057, 3884, 14883, 58508, 235771, 967790, 4037807, 17074475, 73058753, 315803342, 1377445726, 6056134719, 26817483095, 119516734167, 535751271345, 2414304071965, 10932421750492, 49723583969029, 227079111492652, 1040939109111200, 4788357522831785
Offset: 0
a(2) = 2 because at most one node can be colored white.
a(3) = 4 because the only tree is the path graph P_3. If the center node is colored white then both of the ends must be colored black; otherwise zero, one or both of the ends can be colored black. In total there are 4 possibilities.
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EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(u=v=w=[]); for(n=1, n, my(t1=EulerT(v), t2=EulerT(u+v)); u=concat([1], EulerT(u+v+w)); v=concat([0], t2-t1); w=concat([1], t1)); my(g=x*Ser(u+v), guw=x^2*Ser(u)*Ser(w)); Vec(1 + g + (subst(g,x,x^2) - g^2 - 2*guw)/2)}
A339838
Number of rooted bicolored trees on n unlabeled nodes such that black nodes are not adjacent to each other and every white node is adjacent to a black node.
Original entry on oeis.org
1, 2, 4, 10, 27, 75, 221, 662, 2042, 6402, 20407, 65828, 214720, 706600, 2343767, 7826752, 26293468, 88796471, 301290197, 1026595232, 3511246069, 12050780294, 41488523002, 143246116231, 495881545520, 1720771421470, 5984652387281, 20857113949868, 72829214554641, 254762923125929
Offset: 1
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EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(u=v=w=[]); for(n=1, n, my(t1=EulerT(v), t2=EulerT(u+v)); u=concat([1], EulerT(v+w)); v=concat([0], t2-t1); w=concat([1], t1)); u+v}
A340021
Number of bicolored graphs on n unlabeled nodes such that black nodes are not adjacent to each other and every white node is adjacent to a black node.
Original entry on oeis.org
1, 1, 2, 5, 16, 66, 407, 3948, 66781, 2057140, 117820559, 12562407832, 2488441442819, 915216371901462, 625792587599236833, 797474948692631218674, 1899724021357155410243835, 8486672841492724213636009230, 71324140440429733888694354552551, 1131126439181050621704917376323373818
Offset: 0
A049312 counts bicolored graphs where adjacent nodes cannot have the same color.
A000666 counts bicolored graphs where adjacent nodes can have the same color.
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permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2]];
dom[u_, v_] := Product[2^Sum[GCD[u[[i]], v[[j]]], {j, 1, Length[v]}] - 1, {i, 1, Length[u]}];
U[nb_, nw_] := Module[{s = 0}, Do[t = 0; Do[t += permcount[v]*dom[u, v], {v, IntegerPartitions[nb]}]; s += t*permcount[u]*2^edges[u]/nb!, {u, IntegerPartitions[nw]}]; s/nw!];
a[n_] := Sum[U[k, n - k], {k, 0, n}];
Array[a, 20] (* Jean-François Alcover, Jan 07 2021, after Andrew Howroyd *)
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permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
dom(u, v) = {prod(i=1, #u, 2^sum(j=1, #v, gcd(u[i], v[j]))-1)}
U(nb, nw)={my(s=0); forpart(u=nw, my(t=0); forpart(v=nb, t += permcount(v) * dom(u, v)); s += t*permcount(u) * 2^edges(u)/nb!); s/nw!}
a(n)={sum(k=0, n, U(k, n-k))}
Showing 1-4 of 4 results.
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