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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.

%I A194725 #22 Jul 19 2017 20:54:52
%S A194725 1,1,9,97,1145,14289,185193,2467137,33563481,464221105,6507351113,
%T A194725 92236247841,1319640776249,19031570387857,276368559434025,
%U A194725 4037555902072065,59299855337012505,875056238174271345,12967283824008178185,192889769468751321825,2879117809973276680185
%N 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.
%H A194725 Alois P. Heinz, <a href="/A194725/b194725.txt">Table of n, a(n) for n = 0..200</a>
%H A194725 C. Kassel and C. Reutenauer, <a href="https://arxiv.org/abs/1303.3481">Algebraicity of the zeta function associated to a matrix over a free group algebra</a>, arXiv preprint arXiv:1303.3481, 2013
%F A194725 G.f.: 4/5 + 8/(5*(3+5*sqrt(1-16*x))).
%F A194725 a(0) = 1, a(n) = 1/n * Sum_{j=0..n-1} C(2*n,j)*(n-j)*4^j for n>0.
%F A194725 a(n) ~ 2^(4*n+2)/(9*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Aug 13 2013
%F A194725 Recurrence: n*a(n) = (41*n-24)*a(n-1) - 200*(2*n-3)*a(n-2). - _Vaclav Kotesovec_, Aug 13 2013
%e A194725 a(2) = 9: aaaa, aabb, aacc, aadd, aaee, abba, acca, adda, aeea (with 5-ary alphabet {a,b,c,d,e}).
%p A194725 a:= n-> `if`(n=0, 1, add(binomial(2*n, j) *(n-j) *4^j, j=0..n-1) /n):
%p A194725 seq(a(n), n=0..20);
%p A194725 # second Maple program
%p A194725 a:= proc(n) a(n):= `if`(n<3, [1, 1, 9][n+1],
%p A194725        ((41*n-24)*a(n-1) +(600-400*n)*a(n-2))/n)
%p A194725     end:
%p A194725 seq(a(n), n=0..20);  # _Alois P. Heinz_, Oct 30 2012
%t A194725 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 *)
%Y A194725 Column k=5 of A183134.
%K A194725 nonn
%O A194725 0,3
%A A194725 _Alois P. Heinz_, Sep 02 2011