A194725 Number of 5-ary 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, 9, 97, 1145, 14289, 185193, 2467137, 33563481, 464221105, 6507351113, 92236247841, 1319640776249, 19031570387857, 276368559434025, 4037555902072065, 59299855337012505, 875056238174271345, 12967283824008178185, 192889769468751321825, 2879117809973276680185

Offset: 0

Author

Alois P. Heinz, Sep 02 2011

Keywords

• nonn

Examples

			a(2) = 9: aaaa, aabb, aacc, aadd, aaee, abba, acca, adda, aeea (with 5-ary alphabet {a,b,c,d,e}).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• 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=5 of A183134.

Programs

Maple

a:= n-> `if`(n=0, 1, add(binomial(2*n, j) *(n-j) *4^j, j=0..n-1) /n):
seq(a(n), n=0..20);
# second Maple program
a:= proc(n) a(n):= `if`(n<3, [1, 1, 9][n+1],
       ((41*n-24)*a(n-1) +(600-400*n)*a(n-2))/n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 30 2012

Mathematica

FullSimplify[Flatten[{1,Table[4^(2*n+1)*(1/2 (2*n-1))! Hypergeometric2F1[1,1/2+n,2+n,16/25]/(25*Sqrt[Pi]*(n+1)!),{n,1,20}]}]] (* Vaclav Kotesovec, Aug 13 2013 *)

Formula

G.f.: 4/5 + 8/(5*(3+5*sqrt(1-16*x))).

a(0) = 1, a(n) = 1/n * Sum_{j=0..n-1} C(2*n,j)*(n-j)*4^j for n>0.

a(n) ~ 2^(4*n+2)/(9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013

Recurrence: n*a(n) = (41*n-24)*a(n-1) - 200*(2*n-3)*a(n-2). - Vaclav Kotesovec, Aug 13 2013