OFFSET
1,2
COMMENTS
a(n) is the number of distinct matrix products in (A+B+C+D+E)^n where A,B,C and D all commute with each other, but not with E. - Paul D. Hanna and Max Alekseyev, Feb 01 2006
Row sums of Riordan array (1,1/(1-x)^4). - Paul Barry, Feb 02 2006
From Gary W. Adamson, Apr 23 2009: (Start)
Equals the INVERT transform of the tetrahedral series.
a(4) = 69 = (1, 4, 10) dot (19, 5, 1) + 20; = (19 + 20 + 10) + 20. (End)
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, example 16.
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
Index entries for linear recurrences with constant coefficients, signature (5,-6,4,-1).
FORMULA
a(n) = 5*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) = a(n-1)+A055990(n) = A055988(n+1)-A055988(n) = A055989(n+1)-2*A055989(n)+A055989(n-1).
Letting a(0)=1, we have a(n)=sum(u=0, n-1, sum(v=0, u, sum(w=0, v, sum(x=0, w, a(x))))) for n>0. - Benoit Cloitre, Jan 26 2003
a(n) = sum_{k=1..n} binomial(n+3*k-1, n-k). - Vladeta Jovovic, Mar 23 2003
a(n) = sum{k=0..n, binomial(4n-3k-1,k)}. - Paul Barry, Feb 02 2006
G.f.: x/(1-5x+6x^2-4x^3+x^4). - Paul Barry, Feb 02 2006
MATHEMATICA
LinearRecurrence[{5, -6, 4, -1}, {1, 5, 19, 69}, 30] (* Harvey P. Dale, Feb 27 2013 *)
PROG
(Magma) I:=[1, 5, 19, 69]; [n le 4 select I[n] else 5*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Apr 05 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Henry Bottomley, Jun 02 2000
STATUS
approved