A194724 Number of quaternary 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, 7, 58, 523, 4966, 48838, 492724, 5068915, 52955950, 560198962, 5987822380, 64563867454, 701383563388, 7668869344108, 84326618668648, 931894610845123, 10344218506421758, 115280448164645818, 1289346114476360188, 14467472108268263818, 162816535672067515828

Offset: 0

Author

Alois P. Heinz, Sep 02 2011

Keywords

• nonn

Examples

			a(2) = 7: aaaa, aabb, aacc, aadd, abba, acca, adda (with quaternary alphabet {a,b,c,d}).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500
• C. Kassel and C. Reutenauer, Algebraicity of the zeta function associated to a matrix over a free group algebra, arXiv preprint arXiv:1303.3481, 2013

Crossrefs

Column k=4 of A183134.

Programs

Maple

a:= n-> `if`(n=0, 1, add(binomial(2*n, j) *(n-j) *3^j, j=0..n-1)/n):
seq(a(n), n=0..25);
# second Maple program:
a:= proc(n) option remember; `if`(n<3, [1, 1, 7][n+1],
      ((28*n-18)*a(n-1) -(192*n-288)*a(n-2))/n)
    end:
seq(a(n), n=0..30);

Mathematica

CoefficientList[Series[3/4+3/(2(2+4Sqrt[1-12x])),{x,0,30}],x] (* Harvey P. Dale, Sep 30 2012 *)

Formula

G.f.: 3/4 + 3/(2*(2+4*sqrt(1-12*x))).

a(0) = 1, a(n) = 1/n * Sum_{j=0..n-1} C(2*n,j)*(n-j)*3^j for n>0.

a(n) ~ 3 * 12^n / (4 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 07 2014

Conjecture: n*a(n) +2*(-14*n+9)*a(n-1) +96*(2*n-3)*a(n-2)=0. - R. J. Mathar, Mar 14 2015