A383409 Expansion of e.g.f. (exp(x)-1)*(exp(x)-x)*(exp(x)-x^2/2)*(exp(x)-x^3/6).

0, 1, 5, 19, 77, 326, 1406, 5601, 23715, 101092, 431172, 1841357, 7889877, 33924268, 146103678, 628595097, 2695143751, 11495831852, 48733234456, 205252231229, 858955851705, 3573016550756, 14781047390930, 60846099935609, 249385924540907

Offset

0,3

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 do not contain exactly three 3s.

Links

Table of n, a(n) for n=0..24.

Formula

a(n) = 4^n - 3^n - n*(3^(n-1) - 2^(n-1)) - binomial(n,2)*(3^(n-2) - 2^(n-2)) - binomial(n,3)*(3^(n-3) - 2^(n-1) + 3) + binomial(n,4)*(2^(n-2) - 4) + 5*binomial(n,5)*(2^(n-4) - 2) - 60*binomial(n,6) except at n = 6.

Example

a(3)=19 since the strings are: 011 (3 of this type), 033 (3 of this type), 002 (3 of this type), 003 (3 of this type), 023 (6 of this type), and 000.

Crossrefs

Cf. A383323.

Sequence in context: A149771 A149772 A370194 * A149773 A363548 A149774

Adjacent sequences: A383406 A383407 A383408 * A383410 A383411 A383412

Keyword

nonn

Author

Enrique Navarrete, Apr 26 2025

Status

approved