0,3

Sum[a=0..n, Sum[b=0..n, Sum[c=0..n, C(n-b-c,a)*C(n-a-c,b)*C(n-a-b,c) ]]].

a(n+1) = number of unique matrix products in (A+B+C+D)^n where commutator [A,B]=[A,D]=[B,D]=0 but D does not commute with A, B, or C. - Paul D. Hanna and Max Alekseyev (maxale(AT)gmail.com), Feb 01 2006

Starting (1, 4, 13,...) = INVERT transform of the triangular series, (1, 3, 6, 10,...). Example: a(5) = 129 = termwise products of (1, 1, 4, 13, 41) and (15, 10, 6, 3, 1) = (15 + 10 + 24 + 39 + 41). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 10 2009]

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 80.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 459

G.f.: (-1+x)^3/(-1+4*x-3*x^2+x^3)

Recurrence: a(n)=4*a(n-1)-3*a(n-2)+a(n-3) for n>=4.

Sum(-1/31*(5*_alpha+3*_alpha^2-6)*_alpha^(-1-n), _alpha=RootOf(-1+4*_Z-3*_Z^2+_Z^3))

For n>0, a(n)=sum(k=0, n-1, sum(i=0, k, sum(j=0, i, a(j)))). - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 26 2003

a(n) = Sum_{k=0..n} binomial(n+2*k-1, n-k). - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 23 2003

spec := [S, {S=Sequence(Prod(Z, Sequence(Z), Sequence(Z), Sequence(Z)))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);

Cf. A001906, A055991.

Trisection of A000930. First differences of A052544.

Sequence in context: A097112 A077284 A070428 this_sequence A049222 A001453 A141364

Adjacent sequences: A052526 A052527 A052528 this_sequence A052530 A052531 A052532

easy,nonn

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 08 2000