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A001677
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Number of series-parallel networks with n edges.
(Formerly M0797 N0302)
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2
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1, 2, 3, 6, 12, 26, 59, 146, 368, 976, 2667, 7482, 21440, 62622, 185637, 557680, 1694256, 5198142, 16086486, 50165218, 157510504, 497607008, 1580800091, 5047337994, 16190223624, 52153429218, 168657986843, 547389492416
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OFFSET
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2,2
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
B. D. H. Tellegen, Geometrical configurations and duality of electrical networks, Philips Technical Review, 5 (1940), 324-330.
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LINKS
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FORMULA
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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.
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EXAMPLE
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a(5) = 24 - (1/2)*(1*10+2*4+4*2+10*1) = 6.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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