OFFSET
1,3
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
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)
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 *)
PROG
(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 */
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Pab Ter (pabrlos(AT)yahoo.com), May 27 2004
STATUS
approved