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A086405 Row T(n,3) of number array A086404.

%I A086405 #58 Aug 24 2024 18:17:34
%S A086405 1,4,18,84,396,1872,8856,41904,198288,938304,4440096,21010752,
%T A086405 99423936,470479104,2226331008,10535111424,49852682496,235905426432,
%U A086405 1116316463616,5282466223104,24996898556928,118286594002944
%N A086405 Row T(n,3) of number array A086404.
%C A086405 Binomial transform of A079935.
%C A086405 Number of nonisomorphic graded posets with 0 of rank n+1, with exactly 2 elements of each rank level above 0.  Here, we do not assume all maximal elements have maximal rank and thus use graded poset to mean: For every element x, all maximal chains among those with x as greatest element have the same finite length. - _David Nacin_, Feb 13 2012
%D A086405 R. Stanley, Enumerative combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.
%H A086405 Vincenzo Librandi, <a href="/A086405/b086405.txt">Table of n, a(n) for n = 0..1000</a>
%H A086405 Wikipedia, <a href="http://en.wikipedia.org/wiki/Graded_poset">Graded Poset</a>
%H A086405 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-6).
%F A086405 G.f.: (1-2*x)/((1-(3-sqrt(3))*x)*(1-(3+sqrt(3))*x)) = (1-2*x)/(1-6*x+6*x^2);
%F A086405 a(n) = (3-sqrt(3))^n*(1/2 - 1/(2*sqrt(3))) + (3 + sqrt(3))^n*(1/2 + 1/(2*sqrt(3))).
%F A086405 E.g.f.: exp(3*x)*(cosh(sqrt(3)*x) + sinh(sqrt(3)*x)/sqrt(3)). - _Paul Barry_, Nov 20 2003
%F A086405 a(n) = Sum_{k=1..floor(n/2)} C(n, 2k)*3^(n-k-1). - _Paul Barry_, Nov 22 2003
%F A086405 a(n) = (((1+sqrt(3))*(3+sqrt(3))^n) - ((1-sqrt(3))*(3-sqrt(3))^n))/sqrt(12). - Al Hakanson (hawkuu(AT)gmail.com), Jun 10 2009
%F A086405 a(n) = Sum_{k=0..n} A117317(n,k)*2^k. - _Philippe Deléham_, Jan 28 2012
%F A086405 a(n) = 6*(a(n-1) - a(n-2)), a(0)=1, a(1)=4. - _David Nacin_, Feb 27 2012
%F A086405 G.f.: (1-2*x)/(1-6*x+6*x^2). - _Colin Barker_, Aug 04 2012
%t A086405 LinearRecurrence[{6, -6}, {1, 4}, 60] (* _David Nacin_, Feb 27 2012 *)
%o A086405 (Python)
%o A086405 def a(n, adict={0:1, 1:4}):
%o A086405     if n in adict:
%o A086405         return adict[n]
%o A086405     adict[n]=6*a(n-1)-6*a(n-2)
%o A086405     return adict[n] # _David Nacin_, Feb 27 2012
%Y A086405 Cf. A079935, A086404.
%K A086405 easy,nonn
%O A086405 0,2
%A A086405 _Paul Barry_, Jul 19 2003