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 A047098 a(n) = 2*binomial(3*n, n) - Sum_{k=0..n} binomial(3*n, k). 9
 1, 2, 8, 38, 196, 1062, 5948, 34120, 199316, 1181126, 7080928, 42860534, 261542752, 1607076200, 9934255472, 61732449648, 385393229460, 2415935640198, 15200964233864, 95962904716402, 607640599286276, 3858198001960438, 24559243585545644, 156692889782067712 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS T(2n,n), array T as in A047089. [Corrected Dec 08 2006] Let B_3^+ denote the semigroup with presentation . Let D=aba be the 'fundamental word'. Then this sequence is also equal to the number of words in B_3^+ equal in B_3^+ to D^n, n >= 0. - Stephen P. Humphries, Jan 20 2004 In the language of Riordan arrays, row sums of (1/(1+x), x/(1+x)^3)^-1, where (1/(1+x), x/(1+x)^3) has general term (-1)^(n-k)*binomial(n+2k, 3k). - Paul Barry, May 09 2005 Hankel transform is 2^n*A051255(n) where A051255 is the Hankel transform of C(3n,n)/(2n+1). - Paul Barry, Jan 21 2007 LINKS Michael De Vlieger, Table of n, a(n) for n = 0..1211 Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021. Christopher R. Cornwell and Stephen P. Humphries, Counting fundamental paths in certain Garside semigroups, Journal of Knot Theory and Its Ramifications, Vol. 17 (2008), No. 02, pp. 191-211. FORMULA G.f. A(x)=y satisfies (8x-1)y^3-y^2+y+1=0. - Michael Somos, Jan 28 2004 Coefficient of x^n in ((1+10x-2x^2+(1-4x)^(3/2))/2)^n. - Michael Somos, Sep 25 2003 a(n) = Sum_{k = 0..n} A109971(k)*2^k; a(0) = 1, a(n) = Sum_{k = 0..n} 2^k*C(3n-k,n-k)*2*k/(3*n-k), n > 0. - Paul Barry, Jan 21 2007 Conjecture: 2*n*(2*n-1)*a(n) +(-71*n^2+112*n-48)*a(n-1) +3*(131*n^2-391*n+296)*a(n-2) -72*(3*n-7)*(3*n-8)*a(n-3)=0. - R. J. Mathar, Nov 30 2012 a(n) = A321957(n) + 2*binomial(3*n, n) - 8^n. - Peter Luschny, Nov 22 2018 The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. - Peter Bala, Mar 05 2022 MAPLE A047098 := n -> 2*binomial(3*n, n)-add(binomial(3*n, k), k=0..n); MATHEMATICA Table[2Binomial[3n, n]-Sum[Binomial[3n, k], {k, 0, n}], {n, 0, 35}] (* Harvey P. Dale, Jul 27 2011 *) PROG (PARI) a(n)=if(n<0, 0, polcoeff((((1+10*x-2*x^2)+(1-4*x)*sqrt(1-4*x+x*O(x^n)))/2)^n, n)) (PARI) a(n)=if(n<0, 0, 2*binomial(3*n, n)-sum(k=0, n, binomial(3*n, k))) CROSSREFS Column k=2 of A213028. Cf. A047089, A047099, A107026, A321957. Sequence in context: A345178 A026939 A291088 * A271934 A364723 A372107 Adjacent sequences: A047095 A047096 A047097 * A047099 A047100 A047101 KEYWORD nonn AUTHOR Clark Kimberling, Aug 15 1998 EXTENSIONS Clark Kimberling, Dec 08 2006, changed "T(3n,2n)" to "T(2n,n)" in the comment line, but observes that some of the other comments seem to apply to the sequence T(3n,2n) rather than to the sequence T(2n,n). Edited by N. J. A. Sloane, Dec 21 2006, replacing the old definition in terms of A047089 by an explicit formula supplied by Benoit Cloitre, Oct 25 2003. STATUS approved

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Last modified August 7 09:14 EDT 2024. Contains 375008 sequences. (Running on oeis4.)