A194723 Number of ternary words either empty or beginning with the first character of the alphabet, that can be built by inserting n doublets into the initially empty word.

1, 1, 5, 29, 181, 1181, 7941, 54573, 381333, 2699837, 19319845, 139480397, 1014536117, 7426790749, 54669443141, 404388938349, 3004139083221, 22402851226749, 167640057210981, 1258340276153229, 9471952718661621, 71481616200910749, 540715584181142661

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

C. Kassel and C. Reutenauer, Algebraicity of the zeta function associated to a matrix over a free group algebra, arXiv preprint arXiv:1303.3481 [math.CO], 2013-2014.

Formula

G.f.: 2/3 + 4/(3*(1+3*sqrt(1-8*x))).

a(0) = 1, a(n) = 1/n * Sum_{j=0..n-1} C(2*n,j)*(n-j)*2^j for n>0.

D-finite with recurrence: n*a(n) = (17*n-12)*a(n-1) - 36*(2*n-3)*a(n-2). - Vaclav Kotesovec, Oct 20 2012

a(n) ~ 2^(3*n+1)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 20 2012

G.f.: 2-4/( Q(0) + 3), where Q(k) = 1 + 8*x*(4*k+1)/( 4*k+2 - 8*x*(4*k+2)*(4*k+3)/( 8*x*(4*k+3) + 4*(k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 20 2013

From Karol A. Penson, Jul 13 2015: (Start)

Special values of the hypergeometric function 2F1, in Maple notation:

a(n+1) = (16/9)*8^n*GAMMA(n+3/2)*hypergeom([1, n+3/2], [n+3],8/9)/(sqrt(Pi)*(n+2)!), n=0,1,... .

Integral representation as the n-th moment of a positive function W(x) = sqrt((8-x)*x)*(1/(9-x))/(2*Pi) on (0,8): a(n+1) = int(x^n*W(x),x=0..8), n=0,1,... . This representation is unique as W(x) is the solution of the Hausdorff moment problem. (End)

a(n) = 2^(n+1)*binomial(2*n,n)*hypergeom([2,1-n],[n+2],-2)/(n+1) - 3^(2*n-1) for n>=1. - Peter Luschny, Apr 07 2018

Example

a(2) = 5: aaaa, aabb, aacc, abba, acca (with ternary alphabet {a,b,c}).

Maple

a:= n-> `if`(n=0, 1, add(binomial(2*n, j) *(n-j) *2^j, j=0..n-1)/n):
seq(a(n), n=0..25);

Mathematica

CoefficientList[Series[2/3+4/(3*(1+3*Sqrt[1-8*x])), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)
a[n_] := 2^(n+1) CatalanNumber[n] Hypergeometric2F1[2, 1-n, n+2, -2] - 3^(2n - 1);
Table[If[n == 0, 1, a[n]], {n, 0, 22}] (* Peter Luschny, Apr 08 2018 *)

Prog

(PARI) a(n) = if (n==0, 1, sum(j=0, n-1, binomial(2*n, j)*(n-j)*2^j)/n); \\ Michel Marcus, Apr 07 2018
(PARI) x='x+O('x^99); Vec(4/(3*(1+3*(1-8*x)^(1/2)))+2/3) \\ Altug Alkan, Apr 07 2018
(Magma) [1] cat [&+[(Binomial(2*n, k)*(n-k)*2^k)/n: k in [0..n]]: n in [1..25]]; // Vincenzo Librandi, Apr 08 2018

Crossrefs

Column k=3 of A183134.

Cf. A194726.

Sequence in context: A139174 A290117 A153296 * A190917 A153391 A175891

Adjacent sequences: A194720 A194721 A194722 * A194724 A194725 A194726

Keyword

nonn

Author

Alois P. Heinz, Sep 02 2011

Status

approved