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

0, 1, 1, 6, 12, 40, 120, 224, 896, 1152, 5760, 5632, 33792, 26624, 186368, 122880, 983040, 557056, 5013504, 2490368, 24903680, 11010048, 121110528, 48234496, 578813952, 209715200, 2726297600, 905969664, 12683575296, 3892314112, 58384711680, 16642998272, 266287972352, 70866960384

Offset: 0

Author

Enrique Navarrete, Jul 04 2025

Keywords

• nonn
• easy

Comments

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.

A229580 and (1/3)*A229679 interleaved.

Examples

			a(5) = 40 since the strings are the 5 permutations of 10000, the 5 permutations of 12222 and the 30 permutations of 10022.
a(6) = 120 since the strings are the 15 permutations of 110000, the 90 permutations of 110022, and the 15 permutations of 112222.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (0,12,0,-48,0,64).

Crossrefs

Cf. A229580, A229679, A385083.

Formula

a(n) = n*2^(n-2) for n odd, n >= 3.

a(n) = binomial(n,2)*2^(n-3) for n even, n >= 4.

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