1,2

a(n) = number of unique 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 (maxale(AT)gmail.com), Feb 01 2006

Row sums of Riordan array (1,1/(1-x)^4). - Paul Barry (pbarry(AT)wit.ie), Feb 02 2006

Quadrisection of A003269: a(n)=A003269(4n-1); - Paul Barry (pbarry(AT)wit.ie), Feb 02 2006

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), 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)

a(n) = 5a(n-1)-6a(n-2)+4a(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 (benoit7848c(AT)orange.fr), Jan 26 2003

a(n) = Sum_{k=1..n} binomial(n+3*k-1, n-k). - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 23 2003

a(n)=sum{k=0..n, binomial(4n-3k-1,k)}; - Paul Barry (pbarry(AT)wit.ie), Feb 02 2006

G.f.: (1-x)^4/(1-5x+6x^2-4x^3+x^4); - Paul Barry (pbarry(AT)wit.ie), Feb 02 2006

Cf. A055988, A055989, A055990 for the other differences of a(n). See A000079, A001906, A052529 for examples of sequences which are respectively their own first, second and third differences.

Sequence in context: A070857 A143954 A047145 this_sequence A030662 A149758 A026590

Adjacent sequences: A055988 A055989 A055990 this_sequence A055992 A055993 A055994

nonn

Henry Bottomley (se16(AT)btinternet.com), Jun 02 2000

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 05 2000