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A258490 Number of words of length 2n such that all letters of the ternary alphabet occur at least once and are introduced in ascending order and which can be built by repeatedly inserting doublets into the initially empty word.

%I A258490 #14 May 18 2018 17:02:04
%S A258490 5,56,465,3509,25571,184232,1325609,9567545,69387483,505915981,
%T A258490 3708195075,27314663271,202116910415,1501769001416,11200258810265,
%U A258490 83815491037841,629152465444715,4735907436066401,35740538971518155,270356740041089471,2049510329494271615
%N A258490 Number of words of length 2n such that all letters of the ternary alphabet occur at least once and are introduced in ascending order and which can be built by repeatedly inserting doublets into the initially empty word.
%H A258490 Alois P. Heinz, <a href="/A258490/b258490.txt">Table of n, a(n) for n = 3..1000</a>
%F A258490 a(n) ~ 8^n / (sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jun 01 2015
%F A258490 Conjecture: 4*n*(n-1)*(46829*n-161203)*a(n) -(n-1)*(4865671*n^2-22433759*n+19821114)*a(n-1) +6*(7756949*n^3-53792553*n^2+117956226*n-84118712)*a(n-2) +(-200071007*n^3+1677158106*n^2-4623144589*n+4201946850)*a(n-3) +2*(2*n-7)*(93171685*n^2-585009841*n+881711802)*a(n-4) -72*(2*n-7)*(2*n-9)*(744719*n-1901876)*a(n-5)=0. - _R. J. Mathar_, Aug 07 2015
%e A258490 a(3) = 5: aabbcc, aabccb, abbacc, abbcca, abccba.
%p A258490 A:= proc(n, k) option remember; `if`(n=0, 1, k/n*
%p A258490       add(binomial(2*n, j)*(n-j)*(k-1)^j, j=0..n-1))
%p A258490     end:
%p A258490 T:= (n, k)-> add((-1)^i*A(n, k-i)/(i!*(k-i)!), i=0..k):
%p A258490 a:= n-> T(n, 3):
%p A258490 seq(a(n), n=3..25);
%t A258490 A[n_, k_] := A[n, k] = If[n == 0, 1, k/n*Sum[Binomial[2*n, j]*(n - j)*If[j == 0, 1, (k - 1)^j], {j, 0, n - 1}]];
%t A258490 T[n_, k_] := Sum[(-1)^i*A[n, k - i]/(i!*(k - i)!), {i, 0, k}];
%t A258490 a[n_] := T[n, 3];
%t A258490 Table[a[n], {n, 3, 25}] (* _Jean-François Alcover_, May 18 2018, translated from Maple *)
%Y A258490 Column k=3 of A256117.
%K A258490 nonn
%O A258490 3,1
%A A258490 _Alois P. Heinz_, May 31 2015