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 A047099 a(n) = A047098(n)/2. 5
 1, 4, 19, 98, 531, 2974, 17060, 99658, 590563, 3540464, 21430267, 130771376, 803538100, 4967127736, 30866224824, 192696614730, 1207967820099, 7600482116932, 47981452358201, 303820299643138, 1929099000980219, 12279621792772822, 78346444891033856 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A066380, A005809. Column k=2 of A213027. Cf. A001764. Sequence in context: A301417 A025573 A006194 * A211855 A327115 A370024 Adjacent sequences: A047096 A047097 A047098 * A047100 A047101 A047102 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

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Last modified August 3 10:11 EDT 2024. Contains 374885 sequences. (Running on oeis4.)