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

1, 10, 55, 385, 2530, 17017, 113641, 760804, 5089282, 34053437, 227837533, 1524414737, 10199443436, 68241935348, 456589252304, 3054922560820, 20439707165252, 136756870048981, 915005341022187, 6122067418010887, 40961191948244094, 274060890253820561

Offset: 0

Author

Jacques Haubrich (jhaubrich(AT)freeler.nl)

Keywords

• nonn
• easy

Comments

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

References

• J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
• J. Haubrich, Multinacci Rijen [Multinacci sequences], Euclides (Netherlands), Vol. 74, Issue 4, 1998, pp. 131-133.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
• G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.
• Index entries for linear recurrences with constant coefficients, signature (5,15,-20,-35,21,28,-8,-9,1,1).

Crossrefs

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

Programs

Mathematica

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 *)

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)

Formula

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]

Extensions

More terms from Benoit Cloitre, Sep 29 2002

a(20)-a(21) from Vincenzo Librandi, Oct 19 2013