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URL: https://oeis.org/A047098

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A047098
a(n) = 2*binomial(3*n, n) - Sum_{k=0..n} binomial(3*n, k).
13
1, 2, 8, 38, 196, 1062, 5948, 34120, 199316, 1181126, 7080928, 42860534, 261542752, 1607076200, 9934255472, 61732449648, 385393229460, 2415935640198, 15200964233864, 95962904716402, 607640599286276, 3858198001960438, 24559243585545644, 156692889782067712
OFFSET
0,2
COMMENTS
T(2n,n), array T as in A047089. [Corrected Dec 08 2006]
Let B_3^+ denote the semigroup with presentation <a,b | aba=bab>. 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
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
Conjecture verified using the differential equation 6*x + 3 + (-144*x^2 + 114*x - 7)*y + (-1296*x^3 + 792*x^2 - 101*x + 2)*y' + (-648*x^4 + 393*x^3 - 71*x^2 + 4*x)*y'' = 0 satisfied by the g.f. y(x). - Robert Israel, Mar 13 2026
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
From Seiichi Manyama, Nov 08 2025: (Start)
G.f.: g/(2-g) where g = 1+x*g^3 is the g.f. of A001764.
a(n) = (1/(3*n+1)) * Sum_{k=0..n} (3*k+1) * binomial(3*n+1,n-k). (End)
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. A001764.
Sequence in context: A345178 A026939 A291088 * A394113 A271934 A364723
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