A342379 Expansion of e.g.f. (exp(x)-1)*(exp(x) - x^3/6 - x^2/2 - x - 1).

0, 0, 0, 0, 0, 5, 21, 63, 162, 381, 847, 1815, 3796, 7813, 15913, 32191, 64838, 130237, 261155, 523127, 1047224, 2095589, 4192509, 8386559, 16774890, 33551805, 67105911, 134214423, 268431772, 536866821, 1073737297, 2147478655, 4294961806, 8589928573, 17179862603

Offset: 0

Author

Enrique Navarrete, Mar 09 2021

Keywords

• nonn
• easy

Comments

a(n) is the number of binary strings of length n that contain at least four 0's but not all digits are 0.

a(n) is also the number of proper subsets with at least four elements of an n-element set.

Examples

			a(7) = 63 since the strings are the 35 permutations of 0000111, the 21 permutations of 0000011, and the 7 permutations of 0000001.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-9,2).

Crossrefs

Cf. A342352, A002663.

Formula

a(n) = 2^n - Sum_{i={0..3,n}} binomial(n,i).

G.f.: x^5*(2*x^3-7*x^2+9*x-5)/((2*x-1)*(x-1)^4). - Alois P. Heinz, Mar 09 2021