|
|
A058385
|
|
Number of essentially parallel series-parallel networks with n unlabeled edges, multiple edges not allowed.
|
|
3
|
|
|
0, 1, 0, 1, 2, 4, 9, 20, 47, 112, 274, 678, 1709, 4346, 11176, 28966, 75656, 198814, 525496, 1395758, 3723986, 9975314, 26817655, 72332320, 195679137, 530814386, 1443556739, 3934880554, 10748839215, 29420919456, 80678144437, 221618678694
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,5
|
|
LINKS
|
|
|
FORMULA
|
G.f. satisfies 1 - x + x^2 + 2*A(x) = Product_{j>=1} (1-x^j)^(-a(j)).
|
|
MAPLE
|
Q := x; q[1] := 1; for d from 1 to 40 do q[d+1] := c; Q := Q+c*x^(d+1); t0 := mul((1-x^j)^(-q[j]), j=1..d+1); t01 := series(t0, x, d+2); t05 := series(2*Q +1-x+x^2 -t01, x, d+2); t1 := coeff(t05, x, d+1); t2 := solve(t1, c); q[d+1] := t2; Q := subs(c=t2, Q); Q := series(Q, x, d+2); od: A058385 := n->coeff(Q, x, n);
|
|
MATHEMATICA
|
max = 31; f[x_] := Sum[a[k]*x^k, {k, 0, max}]; a[0] = 0; a[1] = 1; a[2] = 0; a[3] = 1; se = Series[ 1 - x + x^2 + 2*f[x] - Product[(1 - x^j)^(-a[j]), {j, 1, max}], {x, 0, max}]; sol = Solve[ Thread[ CoefficientList[ se, x] == 0]]; A058385 = Table[a[n], {n, 0, max}] /. First[sol] (* Jean-François Alcover, Dec 27 2011, after g.f. *)
terms = 32; A[_] = 0; Do[A[x_] = (1/2)*(-1 + x - x^2 + Product[(1 - x^j)^(-Ceiling[Coefficient[A[x], x, j]]), {j, 1, terms}]) + O[x]^ terms // Normal, 4*terms]; CoefficientList[A[x] + O[x]^terms, x] (* Jean-François Alcover, Jan 10 2018 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy,nice
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|