This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A342380 #17 Jun 11 2024 16:50:58
%S A342380 0,0,0,0,0,0,6,28,92,255,637,1485,3301,7098,14912,30826,63018,127857,
%T A342380 258095,519251,1042379,2089604,4185194,8377704,16764264,33539155,
%U A342380 67090961,134196873,268411297,536843070,1073709892,2147447190,4294925846,8589887653,17179816227
%N A342380 Expansion of e.g.f. (exp(x)-1)*(exp(x) - x^4/24 - x^3/6 - x^2/2 - x - 1).
%C A342380 a(n) is the number of binary strings of length n that contain at least five 0's but not all digits are 0.
%C A342380 a(n) is also the number of proper subsets with at least five elements of an n-element set.
%H A342380 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (7,-20,30,-25,11,-2).
%F A342380 a(n) = 2^n - Sum_{i={0..4,n}} binomial(n,i).
%F A342380 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
%e A342380 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.
%t A342380 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 *)
%Y A342380 Cf. A342379, A002664.
%K A342380 nonn,easy
%O A342380 0,7
%A A342380 _Enrique Navarrete_, Mar 09 2021