A086194 Number of unrooted steric quartic trees with n (unlabeled) nodes and possessing a centroid; number of n carbon alkanes C(n)H(2n +2) with a centroid when stereoisomers are regarded as different.

1, 0, 1, 1, 3, 2, 11, 9, 55, 70, 345, 494, 2412, 3788, 18127, 30799, 143255, 256353, 1173770, 2190163, 9892302, 19130814, 85289390, 169923748, 749329719, 1531701274, 6688893605, 13984116304, 60526543480, 129073842978

Offset: 1

Author

Steve Strand (snstrand(AT)comcast.net), Aug 28 2003

Keywords

• nonn

Comments

The degree of each node is <= 4.

A centroid is a node with less than n/2 nodes in each of the incident subtrees, where n is the number of nodes in the tree. If a centroid exists it is unique.

Regarding stereoisomers as different means that only the alternating group A_4 acts at each node, not the full symmetric group S_4. See A010372 for the analogous sequence when stereoisomers are not counted as different.

Links

• Index entries for sequences related to rooted trees
• Index entries for sequences related to trees

Crossrefs

Cf. A000598, A000602, A010732, A010733, A000625, A000628, A086200.

For even n A000628(n) = a(n) + A086200(n/2), for odd n A000628(n) = a(n), since every tree has either a centroid or a bicentroid but not both.

Programs

Mathematica

c[0] = 1; f[x_, m_] := Sum[c[k] x^k, {k, 0, m}]; coes[m_] := CoefficientList[Series[f[x, m] - 1 - (x*(f[x, m]^3 + 2*f[x^3, m])/3), {x, 0, m}], x] // Rest; r[x_, m_] := r[x, m] = (f[x, m] /. Solve[Thread[coes[m] == 0]] // First); b[m_] := CoefficientList[(1/12)*(r[x, m]^4 + 3*r[x^2, m]^2 + 8*r[x, m]*r[x^3, m]), x]; a[1]=1; a[2]=0; a[n_] := b[Quotient[n-1, 2]][[n]]; Table[Print["a(", n, ") = ", an = a[n]]; an, {n, 1, 30}] (* Jean-François Alcover, Dec 29 2014 *)

Formula

Let r(x) = g.f. A(x) for A000625 truncated after the x^n term (x^0 through x^n terms only). Then coefficients of x^(2n) and x^(2n+1) in [r(x)^4 + 8 r(x^3) r(x) + 3 r(x^2)^2]/12 are terms 2n+1 and 2n+2 in current sequence..