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.

5, 56, 465, 3509, 25571, 184232, 1325609, 9567545, 69387483, 505915981, 3708195075, 27314663271, 202116910415, 1501769001416, 11200258810265, 83815491037841, 629152465444715, 4735907436066401, 35740538971518155, 270356740041089471, 2049510329494271615

Offset: 3

Author

Alois P. Heinz, May 31 2015

Keywords

• nonn

Examples

			a(3) = 5: aabbcc, aabccb, abbacc, abbcca, abccba.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..1000

Crossrefs

Column k=3 of A256117.

Programs

Maple

A:= proc(n, k) option remember; `if`(n=0, 1, k/n*
      add(binomial(2*n, j)*(n-j)*(k-1)^j, j=0..n-1))
    end:
T:= (n, k)-> add((-1)^i*A(n, k-i)/(i!*(k-i)!), i=0..k):
a:= n-> T(n, 3):
seq(a(n), n=3..25);

Mathematica

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[n_, k_] := Sum[(-1)^i*A[n, k - i]/(i!*(k - i)!), {i, 0, k}];
a[n_] := T[n, 3];
Table[a[n], {n, 3, 25}] (* Jean-François Alcover, May 18 2018, translated from Maple *)

Formula

a(n) ~ 8^n / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 01 2015

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