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%I #8 Feb 01 2013 06:47:43
%S 0,0,0,3,14,195,2059,31150,489012,9073638,183490118,4135560660,
%T 101421574440,2706766547628,77860733488732,2405136817507216,
%U 79353915366944784,2786110796782734528,103703080088989729280,4079350129335095498048
%N Total number of interior nodes in all essentially parallel series-parallel networks with n labeled edges, multiple edges not allowed.
%D J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226 (the sequence I_Q(n)*Q_pi).
%H <a href="/index/Mo#Moon87">Index entries for sequences mentioned in Moon (1987)</a>
%F Let Q, R = Q-log(1+x), V=Q+R be the e.g.f.'s for A058379, A058380, A058381 resp. E.g.f.'s for A058475, A058406, A058388 are E_V = (V*Q-R)/(1-V), E_R = E_V/(1+V), E_Q = (E_V+V)/(1+V)-Q.
%t max = 19; q = CoefficientList[ InverseSeries[ Series[-1 + E^(1 + 2*a - E^a), {a, 0, max}], x], x]*Table[x^k, {k, 0, max}] // Total; r = q - Log[1 + x]; v = q + r; ev = (v*q - r)/(1 - v); eq = (ev + v)/(1 + v) - q; CoefficientList[ Series[eq, {x, 0, max}], x]*Range[0, max]! (* _Jean-François Alcover_, Feb 01 2013 *)
%K nonn,easy,nice
%O 0,4
%A _N. J. A. Sloane_, Dec 20 2000
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