A331574 a(n) is the number of subsets of {1..n} that contain 3 even and 3 odd numbers.

0, 0, 0, 0, 0, 0, 1, 4, 16, 40, 100, 200, 400, 700, 1225, 1960, 3136, 4704, 7056, 10080, 14400, 19800, 27225, 36300, 48400, 62920, 81796, 104104, 132496, 165620, 207025, 254800, 313600, 380800, 462400, 554880, 665856, 790704, 938961, 1104660, 1299600, 1516200, 1768900, 2048200, 2371600

Offset: 0

Author

Enrique Navarrete, Jan 20 2020

Keywords

• nonn
• easy

Examples

			a(7) = 4 and the 4 subsets are {1,2,3,4,5,6}, {1,2,3,4,6,7}, {1,2,4,5,6,7}, {2,3,4,5,6,7}.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (2,4,-10,-5,20,0,-20,5,10,-4,-2,1).

Crossrefs

Cf. A028723 (2 even and 2 odd numbers).

Programs

Magma

[IsEven(n) select Binomial((n div 2),3)^2 else Binomial((n-1) div 2,3)*Binomial((n+1) div 2,3): n in [0..45]]; //  Marius A. Burtea, Jan 21 2020

Maple

a:= n-> ((b, q)-> b(q, 3)*b(n-q, 3))(binomial, iquo(n, 2)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 30 2020

Mathematica

a[n_] := If[OddQ[n], Binomial[(n - 1)/2, 3]*Binomial[(n + 1)/2, 3], Binomial[n/2, 3]^2]; Array[a, 45, 0] (* Amiram Eldar, Jan 21 2020 *)
LinearRecurrence[{2,4,-10,-5,20,0,-20,5,10,-4,-2,1},{0,0,0,0,0,0,1,4,16,40,100,200},50] (* Harvey P. Dale, Dec 17 2022 *)

PARI

concat([0,0,0,0,0,0], Vec(x^6*(1 + 2*x + 4*x^2 + 2*x^3 + x^4) / ((1 - x)^7*(1 + x)^5) + O(x^40))) \\ Colin Barker, Jan 21 2020

Formula

a(n) = binomial(n/2,3)^2, n even;

a(n) = binomial((n-1)/2,3)*binomial((n+1)/2,3), n odd.

From Colin Barker, Jan 21 2020: (Start)

G.f.: x^6*(1 + 2*x + 4*x^2 + 2*x^3 + x^4) / ((1 - x)^7*(1 + x)^5).

a(n) = 2*a(n-1) + 4*a(n-2) - 10*a(n-3) - 5*a(n-4) + 20*a(n-5) - 20*a(n-7) + 5*a(n-8) + 10*a(n-9) - 4*a(n-10) - 2*a(n-11) + a(n-12) for n>11.

(End)

E.g.f.: (cosh(x)-sinh(x))*(45+36*x+18*x^2+6*x^3+3*x^4+(-45+54*x-36*x^2+18*x^3-9*x^4+6*x^5+2*x^6)*(cosh(2*x)+sinh(2*x)))/4608. - Stefano Spezia, Jan 27 2020