A007137 - OEIS

%I M4734 #56 Dec 26 2018 08:43:28

%S 1,10,98,982,10062,105024,1112757,11934910,129307100,1412855500,

%T 15548498902,172168201088,1916619748084,21436209373224,

%U 240741065193282,2713584138389838,30687358107371442,348061628432108352

%N Number of rooted maps with n edges on the projective plane.

%D E. R. Canfield, Calculating the number of rooted maps on a surface, Congr. Numerantium, 76 (1990), 21-34.

%D David M. Jackson  and Terry I. Visentin, An Atlas of the Smaller Maps in Orientable and Nonorientable Surfaces, Chapman & Hall/CRC, circa 2000. See page 227.

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

%H T. D. Noe, <a href="/A007137/b007137.txt">Table of n, a(n) for n = 1..100</a>

%H E. A. Bender, E. R. Canfield and R. W. Robinson, <a href="http://dx.doi.org/10.4153/CMB-1988-039-4">The enumeration of maps on the torus and the projective plane</a>, Canad. Math. Bull., 31 (1988), 257-271; see p. 270.

%H Guillaume Chapuy, Maciej Dołęga, <a href="https://arxiv.org/abs/1501.06942">A bijection for rooted maps on general surfaces</a>, arXiv:1501.06942 [math.CO], 2016; see corollary 4.5.

%H Valery A. Liskovets, <a href="http://dx.doi.org/10.1016/0012-365X(94)00347-L">A reductive technique for enumerating non-isomorphic planar maps</a>, Discrete Math. 156 (1996), no. 1-3, 197--217. MR1405018 (97f:05087). - _N. J. A. Sloane_, Jun 03 2012

%F From Pab Ter (pabrlos2(AT)yahoo.com), Nov 07 2005: (Start)

%F G.f.: ((2*R+1)/3-sqrt(R*(R+2)/3))/(2*x) where R=sqrt(1-12*x);

%F a(n) ~ sqrt(3/2)*12^n/(n^(5/4)*GAMMA(3/4)). (End)

%F From _Gheorghe Coserea_, Dec 26 2018: (Start)

%F a(n) = (2/(n+1)) * Sum_{k=0..n-1} binomial(2*n, k) * 3^k * A002426(n-k).

%F G.f. y=A(x) satisfies:

%F 0 = 9*x^3*y^4 - 6*x^2*y^3 + 2*x*(21*x - 1)*y^2 + (10*x - 1)*y + x.

%F 0 = x*(4*x + 1)*(12*x - 1)^3*y'''' + 4*(132*x^2 + 19*x - 1)*(12*x - 1)^2*y''' + 12*(1476*x^2 + 60*x - 11)*(12*x - 1)*y'' + 72*(2016*x^2 - 117*x - 4)*y' + 648*(16*x - 1)*y.

%F (End)

%p R:=sqrt(1-12*x): seq(coeff(convert(series(((2*R+1)/3-sqrt(R*(R+2)/3))/(2*x),x,50),polynom),x,n),n=1..25); # Pab Ter, Nov 07 2005

%t With[{r=Sqrt[1-12x]},Rest[CoefficientList[Series[((2r+1)/3-Sqrt[r (r+2)/3])/ (2x),{x,0,20}],x]]](* _Harvey P. Dale_, Mar 02 2018 *)

%o (PARI)

%o seq(N) = {

%o   my(x = 'x + O('x^(N+2)), r=sqrt(1-12*x));

%o   Vec(((2*r+1)/3 - sqrt(r*(r+2)/3))/(2*x));

%o };

%o seq(18)

%o \\ test: y = 'x*Ser(seq(300),'x); 0 == 9*x^3*y^4 - 6*x^2*y^3 + 2*x*(21*x - 1)*y^2 + (10*x - 1)*y + x

%o \\ _Gheorghe Coserea_, Jul 07 2018

%o (PARI)

%o b(n) = sum(k=0, n\2, n!/(k!^2 * (n - 2*k)!)); \\ A002426

%o a(n) = 2*sum(k=0, n-1, binomial(2*n, k) * 3^k * b(n-k))/(n+1);

%o vector(18, n, a(n)) \\ _Gheorghe Coserea_, Dec 26 2018

%Y Cf. A006300.

%Y A column of A267180.

%K nonn,nice

%O 1,2

%A _N. J. A. Sloane_

%E Reference gives 20 terms

%E Description corrected May 15 1997, thanks to Jean-Francois Beraud

%E More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 07 2005