A342380 Expansion of e.g.f. (exp(x)-1)*(exp(x) - x^4/24 - x^3/6 - x^2/2 - x - 1).

0, 0, 0, 0, 0, 0, 6, 28, 92, 255, 637, 1485, 3301, 7098, 14912, 30826, 63018, 127857, 258095, 519251, 1042379, 2089604, 4185194, 8377704, 16764264, 33539155, 67090961, 134196873, 268411297, 536843070, 1073709892, 2147447190, 4294925846, 8589887653, 17179816227

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 five 0's but not all digits are 0.

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

Examples

			a(9) = 255 since the strings are the 126 permutations of 000001111, the 84 permutations of 000000111, the 36 permutations of 000000011, and the 9 permutations of 000000001.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (7,-20,30,-25,11,-2).

Crossrefs

Cf. A342379, A002664.

Programs

Mathematica

LinearRecurrence[{7,-20,30,-25,11,-2},{0,0,0,0,0,0,6,28,92,255,637},40] (* Harvey P. Dale, Jun 11 2024 *)

Formula

a(n) = 2^n - Sum_{i={0..4,n}} binomial(n,i).

G.f.: x^6*(2*x^4-9*x^3+16*x^2-14*x+6)/((2*x-1)*(x-1)^5). - Alois P. Heinz, Mar 09 2021