Series-Parallel Network

Author

Eric W. Weisstein
January 20, 2011

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GraphTheory/Series-ParallelNetwork.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Series-ParallelNetwork.html.

©2010 Wolfram Research, Inc. except for portions noted otherwise

Terms

(PARI) {a(n)=local(A); if(n<1, 0, A=1/(1-x+x*O(x^n)); for(k=2, n, A/=(1-x^k+x*O(x^n))^polcoeff(A, k)); polcoeff(A, n))} /* Michael Somos Oct 11 2006 */

Let b(1)=1, b(n)=a(n)/2 for n >= 2. Then sequence satisfies Product_{k=1..inf} 1/(1-x^k)^b(k) = 1 + Sum_{k=1..inf} a(k)*x^k.

(1/(1-x))Product[(1/(((1-(x^k)))^(a[k]/2))),{k,2,∞}]==1+Sum[a[k](x^k),{k,∞}]

Product[(1/(((1-(x^k)))^(a[k]/2))),{k,2,∞}]

ser=Series[(1/(1-x))Product[(1/(((1-(x^k)))^(a[k]/2))),{k,2,10}],{x,0,10}];

ser = Series[(1/(1 - x))*Product[1/(1 - x^k)^(a[k]/2), {k, 2, 10}], {x, 0, 10}];

Solve[Thread[Rest[CoefficientList[ser,x]]==Array[a,10]],Array[a,10]][[1]][[All,2]]