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A001677 Number of series-parallel networks with n edges.

%I A001677 M0797 N0302 #32 Jul 08 2025 16:26:52
%S A001677 1,2,3,6,12,26,59,146,368,976,2667,7482,21440,62622,185637,557680,
%T A001677 1694256,5198142,16086486,50165218,157510504,497607008,1580800091,
%U A001677 5047337994,16190223624,52153429218,168657986843,547389492416
%N A001677 Number of series-parallel networks with n edges.
%D A001677 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001677 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A001677 B. D. H. Tellegen, Geometrical configurations and duality of electrical networks, Philips Technical Review, 5 (1940), 324-330.
%H A001677 T. D. Noe, <a href="/A001677/b001677.txt">Table of n, a(n) for n=2..500</a>
%H A001677 R. M. Foster, <a href="https://web.archive.org/web/20171111200235/http://www.mathunion.org/ICM/ICM1950.1/Main/icm1950.1.0646.0650.ocr.pdf">The number of series-parallel networks</a>, Proc. Intern. Congr. Math., Vol. 1, 1950, p. 646.
%F A001677 a(n) = s(n) - (1/2)*Sum_{i=1..n-1} s(i)*s(n-i) - (1/2)*s(n/2), where s() = A000084 and the last term is omitted if n is odd.
%e A001677 a(5) = 24 - (1/2)*(1*10+2*4+4*2+10*1) = 6.
%t A001677 m = 29; ClearAll[a, b, s]; a[1] = 1; a[2] = 2; a[3] = 4; b[1] = 1; b[n_ /; n >= 2] = a[n]/2; ex = Product[ 1/(1-x^k)^b[k], {k, 1, m}] - 1 - Sum[ a[k]*x^k, {k, 1, m}]; coes = CoefficientList[ Series[ ex, {x, 0, m}], x]; sol = Solve[ Thread[ coes == 0]][[1]]; Do[ s[k] = a[k] /. sol, {k, 1, m}]; a[2] = 1; a[3] = 2; a[n_] := s[n] - (1/2)*Sum[ s[i]*s[n-i], {i, 1, n-1}] - If[ OddQ[n], 0, s[n/2]/2]; Table[ a[n], {n, 2, m}] (* _Jean-François Alcover_, Feb 24 2012 *)
%Y A001677 Cf. A058642, A058668.
%K A001677 nonn,nice,easy
%O A001677 2,2
%A A001677 _N. J. A. Sloane_
%E A001677 More terms from _David W. Wilson_, Sep 20 2000