A383323 Expansion of e.g.f. (1+x)*(exp(x)-1)*(exp(x)-x)*(exp(x)-x^2/2).

0, 1, 5, 16, 59, 251, 890, 3270, 12269, 45793, 167360, 596036, 2070755, 7041087, 23517590, 77417074, 251879897, 811815485, 2596707692, 8255064768, 26112370895, 82260512731, 258263585090, 808543518254, 2525239747781, 7870664327961, 24487769002520, 76069664095420, 235979863263419

Offset: 0

Author

Enrique Navarrete, Apr 23 2025

Keywords

• nonn

Comments

a(n) is the number of strings of length n defined on {0, 1, 2, 3} that contain at least one 0, do not contain exactly one 1, do not contain exactly two 2s, and contain either one or no 3s.

Examples

			a(3)=16 since the strings are:  011 (3 of this type), 002, (3 of this type), 003, (3 of this type), 023 (6 of this type) and 000.
		

Crossrefs

Cf. A358341.

Formula

a(n) = 3^(n-1)*(n+3) - 2^(n-3)*(3*binomial(n,3) + 6*binomial(n,2) + 8*n + 8) + 12*binomial(n,4) + 6*binomial(n,3) + 3*binomial(n,2) + n except at n = 3, 4.