OFFSET
1,2
COMMENTS
T(2*n,n)/2, with array T as in A047110.
Also given by a recurrence that features row 3 of the Pascal triangle (Mathematica code): u[0,0]=1; u[n_,k_]/;k<0 || k>n := 0; u[n_,k_]/;0<=k<=n := u[n,k] = u[n-1,k-1] + 3u[n-1,k] + 3u[n-1,k+1] + u[n-1,k+2]; u[n_]:=Sum[u[n,k],{k,0,n}]; Table[u[n],{n,0,10}]. - David Callan, Jul 22 2008
INVERT transform of (1,3,12,55,273,...), the ternary numbers A001764. - David Callan, Nov 21 2011
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..400
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
J.-P. Bultel and S. Giraudo, Combinatorial Hopf algebras from PROs, arXiv preprint arXiv:1406.6903 [math.CO], 2014-2016.
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Noga Alon and Noah Kravitz, Counting Dope Matrices, arXiv:2205.09302 [math.CO], 2022.
FORMULA
a(n) = binomial(3*n, n) - (1/2)*Sum_{k=0..n} binomial(3*n, k). - Vladeta Jovovic, Mar 22 2003
a(n) = A047098(n)/2. - Benoit Cloitre, Jan 28 2004
From Gary W. Adamson, Jul 28 2011: (Start)
a(n) is the upper left term in M^n, where M is the infinite square production matrix as follows:
1, 1, 0, 0, 0, 0, ...
3, 3, 1, 0, 0, 0, ...
3, 3, 3, 1, 0, 0, ...
1, 1, 3, 3, 1, 0, ...
0, 0, 1, 3, 3, 0, ...
0, 0, 0, 1, 3, 0, ...
... (End)
G.f.: x*exp( Sum_{n>=1} A066380*x^n/n ) where A066380(n) = Sum_{k=0..n} binomial(3*n,k). - Paul D. Hanna, Sep 04 2012
G.f.: (F(x)-1)/(2-F(x)), where F(x) is g.f. of A001764. - Vladimir Kruchinin, Jun 13 2014.
a(n) = (1/n)*Sum_{k=1..n} k*C(3*n,n-k). - Vladimir Kruchinin, Oct 03 2022
MAPLE
f := n -> binomial(3*n, n) - (1/2)*add(binomial(3*n, k), k=0..n):
seq(f(n), n=1..20);
MATHEMATICA
Table[Binomial[3 n, n] - Sum[Binomial[3 n, k], {k, 0, n}]/2, {n, 20}] (* Wesley Ivan Hurt, Jun 13 2014 *)
PROG
(PARI) {a(n)=local(A=1+x); A=x*exp(sum(m=1, n+1, sum(j=0, m, binomial(3*m, j))*x^m/m +x*O(x^n))); polcoeff(A, n)} \\ Paul D. Hanna, Sep 04 2012
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Dec 11 1999
EXTENSIONS
Comment revised by Clark Kimberling, Dec 08 2006
Edited by N. J. A. Sloane, Dec 21 2006
STATUS
approved