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A030114 Number of distributive lattices; also number of paths with n turns when light is reflected from 10 glass plates. 1

%I

%S 1,10,55,385,2530,17017,113641,760804,5089282,34053437,227837533,

%T 1524414737,10199443436,68241935348,456589252304,3054922560820,

%U 20439707165252,136756870048981,915005341022187,6122067418010887,40961191948244094,274060890253820561

%N Number of distributive lattices; also number of paths with n turns when light is reflected from 10 glass plates.

%C Let M(10) be the 10 X 10 matrix (0,0,0,1)/(0,0,1,1)/(0,1,1,1)/(1,1,1,1) and let v(10) be the vector (1,1,1,1,1,1,1,1,1); then v(10)*M(10)^n = (x,y,z,t,u,v, w,m,a(n)) - _Benoit Cloitre_, Sep 29 2002

%D J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.

%D J. Haubrich, Multinacci Rijen [Multinacci sequences], Euclides (Netherlands), Vol. 74, Issue 4, 1998, pp. 131-133.

%D G. Kreweras, Les preordres totaux compatibles avec un ordre partiel. Math. Sci. Humaines No. 53 (1976), 5-30.

%H Vincenzo Librandi, <a href="/A030114/b030114.txt">Table of n, a(n) for n = 0..1000</a>

%H J. Berman and P. Koehler, <a href="/A006356/a006356.pdf">Cardinalities of finite distributive lattices</a>, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (5,15,-20,-35,21,28,-8,-9,1,1).

%F G.f.: 1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1)))))))))) = -(x^9 +x^8 -8*x^7 -7*x^6 +21*x^5 +15*x^4 -20*x^3 -10*x^2 +5*x +1)/((x +1)*(x^3 +x^2 -2*x -1)*(x^6 -x^5 -6*x^4 +6*x^3 +8*x^2 -8*x +1)). [_Colin Barker_, Nov 09 2012]

%t CoefficientList[Series[-(x^9 + x^8 - 8 x^7 - 7 x^6 + 21 x^5 + 15 x^4 - 20 x^3 - 10 x^2 + 5 x + 1)/((x + 1) (x^3 + x^2 - 2 x - 1) (x^6 - x^5 - 6 x^4 + 6 x^3 8 x^2 - 8 x + 1)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Oct 19 2013 *)

%o (PARI) k=10; M(k)=matrix(k,k,i,j,if(1-sign(i+j-k),0,1)); v(k)=vector(k,i,1); a(n)=vecmax(v(k)*M(k)^n)

%Y See also A006356-A006359, A025030, A030112-A030116.

%K nonn,easy

%O 0,2

%A Jacques Haubrich (jhaubrich(AT)freeler.nl)

%E More terms from _Benoit Cloitre_, Sep 29 2002

%E a(20)-a(21) from _Vincenzo Librandi_, Oct 19 2013

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Last modified November 21 06:00 EST 2019. Contains 329350 sequences. (Running on oeis4.)