The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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. 8
 1, 1, 5, 29, 181, 1181, 7941, 54573, 381333, 2699837, 19319845, 139480397, 1014536117, 7426790749, 54669443141, 404388938349, 3004139083221, 22402851226749, 167640057210981, 1258340276153229, 9471952718661621, 71481616200910749, 540715584181142661 (list; graph; refs; listen; history; text; internal format)
 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified November 30 17:12 EST 2021. Contains 349424 sequences. (Running on oeis4.)