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

%I A047099 #74 Jun 07 2025 08:17:48
%S A047099 1,4,19,98,531,2974,17060,99658,590563,3540464,21430267,130771376,
%T A047099 803538100,4967127736,30866224824,192696614730,1207967820099,
%U A047099 7600482116932,47981452358201,303820299643138,1929099000980219,12279621792772822,78346444891033856
%N A047099 a(n) = A047098(n)/2.
%C A047099 T(2*n,n)/2, with array T as in A047110.
%C A047099 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
%C A047099 INVERT transform of (1,3,12,55,273,...), the ternary numbers A001764. - _David Callan_, Nov 21 2011
%H A047099 Alois P. Heinz, <a href="/A047099/b047099.txt">Table of n, a(n) for n = 1..400</a>
%H A047099 Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.
%H A047099 J.-P. Bultel and S. Giraudo, <a href="http://arxiv.org/abs/1406.6903">Combinatorial Hopf algebras from PROs</a>, arXiv preprint arXiv:1406.6903 [math.CO], 2014-2016.
%H A047099 Isaac DeJager, Madeleine Naquin, and Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.
%H A047099 Noga Alon and Noah Kravitz, <a href="https://arxiv.org/abs/2205.09302">Counting Dope Matrices</a>, arXiv:2205.09302 [math.CO], 2022.
%F A047099 a(n) = binomial(3*n, n) - (1/2)*Sum_{k=0..n} binomial(3*n, k). - _Vladeta Jovovic_, Mar 22 2003
%F A047099 a(n) = A047098(n)/2. - _Benoit Cloitre_, Jan 28 2004
%F A047099 From _Gary W. Adamson_, Jul 28 2011: (Start)
%F A047099 a(n) is the upper left term in M^n, where M is the infinite square production matrix as follows:
%F A047099   1, 1, 0, 0, 0, 0, ...
%F A047099   3, 3, 1, 0, 0, 0, ...
%F A047099   3, 3, 3, 1, 0, 0, ...
%F A047099   1, 1, 3, 3, 1, 0, ...
%F A047099   0, 0, 1, 3, 3, 0, ...
%F A047099   0, 0, 0, 1, 3, 0, ...
%F A047099   ... (End)
%F A047099 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
%F A047099 G.f.: (F(x)-1)/(2-F(x)), where F(x) is g.f. of A001764. - _Vladimir Kruchinin_, Jun 13 2014.
%F A047099 a(n) = (1/n)*Sum_{k=1..n} k*C(3*n,n-k). - _Vladimir Kruchinin_, Oct 03 2022
%F A047099 From _Paul D. Hanna_, Jun 06 2025: (Start)
%F A047099 G.f. A(x) = Series_Reversion( x*(1 + x)^2 / (1 + 2*x)^3 ).
%F A047099 G.f. satisfies A(x) = x*(1 + 2*A(x))^3 / (1 + A(x))^2.
%F A047099 G.f. satisfies A'(x) = A(x) * (1 + A(x)) * (1 + 2*A(x)) / (x*(1 - A(x))).
%F A047099 (End)
%p A047099 f := n -> binomial(3*n, n) - (1/2)*add(binomial(3*n, k), k=0..n):
%p A047099 seq(f(n), n=1..20);
%t A047099 Table[Binomial[3 n, n] - Sum[Binomial[3 n, k], {k, 0, n}]/2, {n, 20}] (* _Wesley Ivan Hurt_, Jun 13 2014 *)
%o A047099 (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
%Y A047099 Cf. A066380, A005809.
%Y A047099 Column k=2 of A213027.
%Y A047099 Cf. A001764.
%K A047099 nonn
%O A047099 1,2
%A A047099 _Clark Kimberling_, Dec 11 1999
%E A047099 Comment revised by _Clark Kimberling_, Dec 08 2006
%E A047099 Edited by _N. J. A. Sloane_, Dec 21 2006