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A007165 Number of P-graphs with 2n edges.
(Formerly M0873)
4

%I M0873

%S 1,1,2,3,8,14,42,79,252,494,1636,3294,11188,22952,79386,165127,579020,

%T 1217270,4314300,9146746,32697920,69799476,251284292,539464358,

%U 1953579240,4214095612,15336931928,33218794236,121416356108

%N Number of P-graphs with 2n edges.

%D R. C. Read, On the enumeration of a class of plane multigraphs, Aequat. Math., 31 (1986), 47-63.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H R. C. Read, <a href="http://dx.doi.org/10.1007/BF02188172">On the enumeration of a class of plane multigraphs</a>, Aequat. Math. 31 (1986) no 1, 47-63

%F From _Paul D. Hanna_, Dec 30 2011: (Start)

%F G.f. satisfies:

%F (1) A(x) = (1/x)*Series_Reversion(x*(1 + 2*x - x^2)/((1+x)*(1+2*x))).

%F (2) A(x) = (1 + x*A(x))*(1 + 2*x*A(x)) / (1 + 2*x*A(x) - x^2*A(x)^2).

%F (3) A(x) = (1 + x*A(x))/(1 - x^2*A(x)*A(-x)).

%F (4) A(x) = A(-x)/(1 - 2*x*A(-x)).

%F (5) A(x) - A(-x) = 2*x*A(x)*A(-x). (End)

%p bq := proc(q::integer) local m; if q mod 2 = 0 then RETURN(0); else m:=(q-1)/2; RETURN( sum(binomial(q,s)*binomial(s+m-1,m),s=0..q)/(q*2^(m+1)) ); fi; end: H := proc(maxord::integer) local resul,r,B; resul := 0; B := 0; for r from 2 to maxord by 2 do B := B+bq(r/2)*y^r; od : for r from 1 to maxord/2 do resul := resul + B^r; od : RETURN(resul); end: maxord := 60 : Hser := expand(H(maxord)) : for n from 2 to maxord by 2 do simplify(coeftayl(Hser,y=0,n)); od; # _R. J. Mathar_, Apr 24 2006

%t 1/x InverseSeries[ x (1 + 2x - x^2)/((1 + x)(1 + 2x)) + O[x]^30] //

%t CoefficientList[#, x]& (* _Jean-Fran├žois Alcover_, Oct 02 2019 *)

%o (PARI) {a(n)=polcoeff(1/x*serreverse(x*(1+2*x-x^2)/((1+x)*(1+2*x)+x*O(x^n))),n)} /* _Paul D. Hanna_ */

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*A*(3-2*A)+x^2*A^2*(2+A)+x*O(x^n));polcoeff(A,n)} /* _Paul D. Hanna_ */

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=(1+x*A)/(1-x^2*A*subst(A,x,-x+x*O(x^n))));polcoeff(A,n)} /* _Paul D. Hanna_ */

%K nonn

%O 1,3

%A _N. J. A. Sloane_

%E More terms from Pab Ter (pabrlos(AT)yahoo.com), May 27 2004

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Last modified June 1 02:09 EDT 2020. Contains 334758 sequences. (Running on oeis4.)