OFFSET
0,5
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..500 (using data from A058387)
S. R. Finch, Series-parallel networks
S. R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
Ji Li, Combinatorial Logarithm and Point-Determining Cographs, Electronic Journal of Combinatorics, 19 (3) (2012), #P8.
J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226 (the sequence q_n).
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
N. J. A. Sloane, Dec 20 2000
STATUS
approved