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 A331419 #30 Jul 22 2024 12:58:12
%S A331419 0,0,0,0,0,0,8,16,80,160,480,960,2240,4480,8960,17920,32256,64512,
%T A331419 107520,215040,337920,675840,1013760,2027520,2928640,5857280,8200192,
%U A331419 16400384,22364160,44728320,59637760,119275520,155975680,311951360,401080320,802160640,1016070144,2032140288
%N A331419 a(n) is the number of subsets of {1..n} that contain exactly 4 odd numbers.
%C A331419 2*a(n-1) for n > 1 is the number of subsets of {1..n} that contain 4 even numbers. For example, for n=8, 2*a(7)=16 and the 16 subsets are {2,4,6,8}, {1,2,4,6,8}, {2,3,4,6,8}, {2,4,5,6,8}, {2,4,6,7,8}, {1,2,3,4,6,8}, {1,2,4,5,6,8}, {1,2,4,6,7,8}, {2,3,4,5,6,8}, {2,3,4,6,7,8}, {2,4,5,6,7,8}, {1,2,3,4,5,6,8}, {1,2,3,4,6,7,8}, {1,2,4,5,6,7,8}, {2,3,4,5,6,7,8}, {1,2,3,4,5,6,7,8}.
%H A331419 Colin Barker, <a href="/A331419/b331419.txt">Table of n, a(n) for n = 1..1000</a>
%H A331419 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,10,0,-40,0,80,0,-80,0,32).
%F A331419 a(n) = binomial((n+1)/2, 4) * 2^((n-1)/2), n odd;
%F A331419 a(n) = binomial((n/2), 4) * 2^(n/2), n even.
%F A331419 From _Colin Barker_, Jan 18 2020: (Start)
%F A331419 G.f.: 8*x^7*(1 + 2*x) / (1 - 2*x^2)^5.
%F A331419 a(n) = 10*a(n-2) - 40*a(n-4) + 80*a(n-6) - 80*a(n-8) + 32*a(n-10) for n>10. (End)
%F A331419 From _Amiram Eldar_, Mar 24 2022: (Start)
%F A331419 Sum_{n>=7} 1/a(n) = (5-6*log(2))/4.
%F A331419 Sum_{n>=7} (-1)^(n+1)/a(n) = (5-6*log(2))/12. (End)
%e A331419 a(7)=8 and the 8 subsets are {1,3,5,7}, {1,2,3,5,7}, {1,3,4,5,7}, {1,3,5,6,7}, {1,2,3,4,5,7}, {1,2,3,5,6,7}, {1,3,4,5,6,7}, {1,2,3,4,5,6,7}.
%t A331419 a[n_] := If[OddQ[n], Binomial[(n + 1)/2, 4]*2^((n - 1)/2), Binomial[n/2, 4]*2^(n/2)]; Array[a, 38] (* _Amiram Eldar_, Jan 17 2020 *)
%t A331419 LinearRecurrence[{0,10,0,-40,0,80,0,-80,0,32},{0,0,0,0,0,0,8,16,80,160},50] (* _Harvey P. Dale_, Jul 22 2024 *)
%o A331419 (Magma) [IsOdd(n) select Binomial((n+1) div 2, 4)*2^((n-1) div 2) else Binomial((n div 2), 4)*2^(n div 2): n in [1..38]]; // _Marius A. Burtea_, Jan 17 2020
%o A331419 (PARI) concat([0,0,0,0,0,0], Vec(8*x^7*(1 + 2*x) / (1 - 2*x^2)^5 + O(x^40))) \\ _Colin Barker_, Jan 18 2020
%Y A331419 Cf. A330592, A331408.
%K A331419 nonn,easy
%O A331419 1,7
%A A331419 _Enrique Navarrete_, Jan 16 2020