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A385601 Expansion of e.g.f. cosh(x)^2*(x+x^2/2).

%I A385601 #7 Jul 08 2025 18:35:15
%S A385601 0,1,1,6,12,40,120,224,896,1152,5760,5632,33792,26624,186368,122880,
%T A385601 983040,557056,5013504,2490368,24903680,11010048,121110528,48234496,
%U A385601 578813952,209715200,2726297600,905969664,12683575296,3892314112,58384711680,16642998272,266287972352,70866960384
%N A385601 Expansion of e.g.f. cosh(x)^2*(x+x^2/2).
%C A385601 a(n) is the number of ternary strings of length n that contain one or two 1's, an even number of 0's, and an even number of 2's.
%C A385601 A229580 and (1/3)*A229679 interleaved.
%H A385601 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,12,0,-48,0,64).
%F A385601 a(n) = n*2^(n-2) for n odd, n >= 3.
%F A385601 a(n) = binomial(n,2)*2^(n-3) for n even, n >= 4.
%F A385601 G.f.: x*(1 + x - 6*x^2 + 16*x^4 + 24*x^5 - 32*x^6 - 32*x^7)/(1 - 4*x^2)^3. - _Stefano Spezia_, Jul 04 2025
%e A385601 a(5) = 40 since the strings are the 5 permutations of 10000, the 5 permutations of 12222 and the 30 permutations of 10022.
%e A385601 a(6) = 120 since the strings are the 15 permutations of 110000, the 90 permutations of 110022, and the 15 permutations of 112222.
%Y A385601 Cf. A229580, A229679, A385083.
%K A385601 nonn,easy
%O A385601 0,4
%A A385601 _Enrique Navarrete_, Jul 04 2025