A330592 - OEIS

%I #27 Mar 24 2022 08:43:58

%S 0,0,2,4,12,24,48,96,160,320,480,960,1344,2688,3584,7168,9216,18432,

%T 23040,46080,56320,112640,135168,270336,319488,638976,745472,1490944,

%U 1720320,3440640,3932160,7864320,8912896,17825792,20054016,40108032,44826624,89653248

%N a(n) is the number of subsets of {1,2,...,n} that contain exactly two odd numbers.

%C 2*a(n-1) for n>1 is the number of subsets of {1,2,...,n} that contain exactly two even numbers.  For example, for n=5, 2*a(4)=8 and the 8 subsets are {2,4}, {1,2,4}, {2,3,4}, {2,4,5}, {1,2,3,4}, {1,2,4,5}, {2,3,4,5}, {1,2,3,4,5}. - _Enrique Navarrete_, Dec 20 2019

%H Colin Barker, <a href="/A330592/b330592.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,0,-12,0,8).

%F a(n) = binomial((n+1)/2,2) * 2^((n-1)/2), n odd;

%F a(n) = binomial(n/2,2) * 2^(n/2), n even.

%F G.f.: 2*(2*x+1)*x^3/(1-2*x^2)^3.

%F a(n) = 6*a(n-2) - 12*a(n-4) + 8*a(n-6) for n>6. - _Colin Barker_, Dec 20 2019

%F From _Amiram Eldar_, Mar 24 2022: (Start)

%F Sum_{n>=3} 1/a(n) = 3*(1-log(2)).

%F Sum_{n>=3} (-1)^(n+1)/a(n) = 1-log(2). (End)

%e For n=5, a(5)=12 and the 12 subsets are {1,3}, {1,5}, {3,5}, {1,2,3}, {1,2,5}, {1,3,4}, {1,4,5}, {2,3,5}, {3,4,5}, {1,2,3,4}, {1,2,4,5}, {2,3,4,5}.

%t a[n_] := If[OddQ[n], Binomial[(n + 1)/2, 2]*2^((n - 1)/2), Binomial[n/2, 2]*2^(n/2)]; Array[a, 38] (* _Amiram Eldar_, Mar 24 2022 *)

%o (Magma) [IsEven(n) select Binomial(n div 2,2)*2^(n div 2) else Binomial((n+1) div 2,2)*2^((n-1) div 2):n in [1..40]]; // _Marius A. Burtea_, Dec 19 2019

%o (PARI) concat([0,0], Vec(2*x^3*(1 + 2*x) / (1 - 2*x^2)^3 + O(x^40))) \\ _Colin Barker_, Dec 20 2019

%Y Cf. A089822 (with exactly two primes).

%K nonn,easy

%O 1,3

%A _Enrique Navarrete_, Dec 18 2019