A001677 Number of series-parallel networks with n edges. (Formerly M0797 N0302)

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

Offset

2,2

References

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.

Links

T. D. Noe, Table of n, a(n) for n=2..500

R. M. Foster, The number of series-parallel networks, Proc. Intern. Congr. Math., Vol. 1, 1950, p. 646.

Formula

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.

Example

a(5) = 24 - (1/2)*(1*10+2*4+4*2+10*1) = 6.

Mathematica

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 *)

Crossrefs

Cf. A058642, A058668.

Sequence in context: A375099 A086625 A152172 * A373182 A339150 A024422

Adjacent sequences: A001674 A001675 A001676 * A001678 A001679 A001680

Keyword

nonn,nice,easy

Author

N. J. A. Sloane.

Extensions

More terms from David W. Wilson, Sep 20 2000

Status

approved