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A086405 Row T(n,3) of number array A086404. 8

%I #55 May 16 2021 05:52:14

%S 1,4,18,84,396,1872,8856,41904,198288,938304,4440096,21010752,

%T 99423936,470479104,2226331008,10535111424,49852682496,235905426432,

%U 1116316463616,5282466223104,24996898556928,118286594002944

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

%C Binomial transform of A079935.

%C 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 R. Stanley, Enumerative combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.

%H Vincenzo Librandi, <a href="/A086405/b086405.txt">Table of n, a(n) for n = 0..1000</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Graded_poset">Graded Poset</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-6).

%F 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 a(n) = (3-sqrt(3))^n*(1/2 - 1/(2*sqrt(3))) + (3 + sqrt(3))^n*(1/2 + 1/(2*sqrt(3))).

%F E.g.f.: exp(3*x)*(cosh(sqrt(3*x) + sinh(sqrt(3)*x)/sqrt(3)). - _Paul Barry_, Nov 20 2003

%F a(n) = Sum_{k=1..floor(n/2)} C(n, 2k)*3^(n-k-1). - _Paul Barry_, Nov 22 2003

%F 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 a(n) = Sum_{k=0..n} A117317(n,k)*2^k. - _Philippe Deléham_, Jan 28 2012

%F a(n) = 6*(a(n-1) - a(n-2)), a(0)=1, a(1)=4. - _David Nacin_, Feb 27 2012

%F G.f.: (1-2*x)/(1-6*x+6*x^2). - _Colin Barker_, Aug 04 2012

%t LinearRecurrence[{6, -6}, {1, 4}, 60] (* _David Nacin_, Feb 27 2012 *)

%o (Python)

%o def a(n, adict={0:1, 1:4}):

%o if n in adict:

%o return adict[n]

%o adict[n]=6*a(n-1)-6*a(n-2)

%o return adict[n] # _David Nacin_, Feb 27 2012

%Y Cf. A079935, A086404.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Jul 19 2003

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