A007165 Number of P-graphs with 2n edges.

1, 1, 2, 3, 8, 14, 42, 79, 252, 494, 1636, 3294, 11188, 22952, 79386, 165127, 579020, 1217270, 4314300, 9146746, 32697920, 69799476, 251284292, 539464358, 1953579240, 4214095612, 15336931928, 33218794236, 121416356108

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

References

• R. C. Read, On the enumeration of a class of plane multigraphs, Aequat. Math., 31 (1986), 47-63.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• R. C. Read, On the enumeration of a class of plane multigraphs, Aequat. Math. 31 (1986) no 1, 47-63

Programs

Maple

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

Mathematica

1/x InverseSeries[ x (1 + 2x - x^2)/((1 + x)(1 + 2x)) + O[x]^30] //
CoefficientList[#, x]& (* Jean-François Alcover, Oct 02 2019 *)

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 */

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 */

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 */

Formula

From Paul D. Hanna, Dec 30 2011: (Start)

G.f. satisfies:

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

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

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

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

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

Extensions

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