A047099 a(n) = A047098(n)/2.

1, 4, 19, 98, 531, 2974, 17060, 99658, 590563, 3540464, 21430267, 130771376, 803538100, 4967127736, 30866224824, 192696614730, 1207967820099, 7600482116932, 47981452358201, 303820299643138, 1929099000980219, 12279621792772822, 78346444891033856

Offset: 1

Author

Clark Kimberling, Dec 11 1999

Keywords

• nonn

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.

Crossrefs

Cf. A066380, A005809.

Column k=2 of A213027.

Cf. A001764.

Programs

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 *)

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

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

From Paul D. Hanna, Jun 06 2025: (Start)

G.f. A(x) = Series_Reversion( x*(1 + x)^2 / (1 + 2*x)^3 ).

G.f. satisfies A(x) = x*(1 + 2*A(x))^3 / (1 + A(x))^2.

G.f. satisfies A'(x) = A(x) * (1 + A(x)) * (1 + 2*A(x)) / (x*(1 - A(x))).

(End)

Extensions

Comment revised by Clark Kimberling, Dec 08 2006

Edited by N. J. A. Sloane, Dec 21 2006