# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a006416 Showing 1-1 of 1 %I A006416 M4490 #69 Apr 13 2022 13:25:18 %S A006416 1,8,20,38,63,96,138,190,253,328,416,518,635,768,918,1086,1273,1480, %T A006416 1708,1958,2231,2528,2850,3198,3573,3976,4408,4870,5363,5888,6446, %U A006416 7038,7665,8328,9028,9766,10543,11360,12218,13118,14061,15048 %N A006416 Number of loopless rooted planar maps with 3 faces and n vertices and no isthmuses. Also a(n)=T(4,n-3), array T as in A049600. %C A006416 If Y_i (i=1,2,3) are 2-blocks of an n-set X then, for n>=6, a(n-3) is the number of (n-3)-subsets of X intersecting each Y_i (i=1,2,3). - _Milan Janjic_, Nov 09 2007 %C A006416 a(n) is also the number of triangle subgraphs in a complete graph on n+3 vertices, minus 3 non-incident edges, for n > 2. - _Robert H Cowen_, Jun 23 2018 %D A006416 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %H A006416 T. D. Noe, Table of n, a(n) for n=2..1000 %H A006416 Milan Janjic, Two Enumerative Functions %H A006416 Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009. %H A006416 Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992 %H A006416 T. R. S. Walsh, A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259. %H A006416 Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1). %F A006416 G.f.: x^2*(1+4*x-6*x^2+2*x^3)/(1-x)^4. %F A006416 a(n-3) = (1/6)*n^3-(1/2)*n^2-(8/3)*n+6, n=6,7,... - _Milan Janjic_, Nov 09 2007 %F A006416 a(2)=1, a(3)=8, a(4)=20, a(5)=38, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - _Harvey P. Dale_, Aug 25 2013 %F A006416 a(n+2) = Hyper2F1([-3, n], [1], -1). - _Peter Luschny_, Aug 02 2014 %F A006416 a(n) = binomial(n+3, 3) - 3*(n+1). - _Robert H Cowen_, Jun 23 2018 %p A006416 A006416:=(1+4*z-6*z**2+2*z**3)/(z-1)**4; # Conjectured by _Simon Plouffe_ in his 1992 dissertation. %p A006416 a := n -> hypergeom([-3, n-2], [1], -1); %p A006416 seq(round(evalf(a(n),32)), n=2..41); # _Peter Luschny_, Aug 02 2014 %t A006416 f[n_]:=Sum[i+i^2-6,{i,1,n}]/2;Table[f[n],{n,3,5!}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 08 2010 *) %t A006416 CoefficientList[Series[(1+4x-6x^2+2x^3)/(1-x)^4,{x,0,50}],x] (* or *) LinearRecurrence[{4,-6,4,-1},{1,8,20,38},50] (* _Harvey P. Dale_, Aug 25 2013 *) %t A006416 f[n_]:= Binomial[n,3] - 3(n-2); Table[{n,f[n]},{n,5,100}]//TableForm (* _Robert H Cowen_, Jun 23 2018 *) %o A006416 (PARI) Vec((1+4*x-6*x^2+2*x^3)/(1-x)^4 + O(x^40)) \\ _Andrew Howroyd_, Jul 15 2018 %Y A006416 Column k=3 of A342980. %Y A006416 Cf. A049600. %K A006416 nonn,easy,nice %O A006416 2,2 %A A006416 _N. J. A. Sloane_ %E A006416 Name clarified by _Andrew Howroyd_, Apr 01 2021 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE