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A007165
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Number of P-graphs with 2n edges.
(Formerly M0873)
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4
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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
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OFFSET
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1,3
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REFERENCES
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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).
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LINKS
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FORMULA
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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)
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MAPLE
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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
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MATHEMATICA
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1/x InverseSeries[ x (1 + 2x - x^2)/((1 + x)(1 + 2x)) + O[x]^30] //
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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More terms from Pab Ter (pabrlos(AT)yahoo.com), May 27 2004
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STATUS
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approved
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