A331575 a(n) is the number of subsets of {1..n} that contain 4 even and 4 odd numbers.

0, 0, 0, 0, 0, 0, 0, 0, 1, 5, 25, 75, 225, 525, 1225, 2450, 4900, 8820, 15876, 26460, 44100, 69300, 108900, 163350, 245025, 353925, 511225, 715715, 1002001, 1366365, 1863225, 2484300, 3312400, 4331600, 5664400, 7282800, 9363600, 11860560, 15023376, 18779220, 23474025, 28997325

Offset

0,10

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,6,-14,-14,42,14,-70,0,70,-14,-42,14,14,-6,-2,1).

Formula

a(n) = binomial(n/2,4)^2, n even;

a(n) = binomial((n-1)/2,4)*binomial((n+1)/2,4), n odd.

From Colin Barker, Jan 21 2020: (Start)

G.f.: x^8*(1 + 3*x + 9*x^2 + 9*x^3 + 9*x^4 + 3*x^5 + x^6) / ((1 - x)^9*(1 + x)^7).

a(n) = 2*a(n-1) + 6*a(n-2) - 14*a(n-3) - 14*a(n-4) + 42*a(n-5) + 14*a(n-6) - 70*a(n-7) + 70*a(n-9) - 14*a(n-10) - 42*a(n-11) + 14*a(n-12) + 14*a(n-13) - 6*a(n-14) - 2*a(n-15) + a(n-16) for n>15.

(End)

E.g.f.: (cosh(x)-sinh(x))*(1575+1350*x+630*x^2+204*x^3+54*x^4+12*x^5+4*x^6+(-1575+1800*x-1080*x^2+456*x^3-156*x^4+48*x^5-16*x^6+8*x^7+2*x^8)*(cosh(2*x)+sinh(2*x)))/294912. - Stefano Spezia, Jan 27 2020

Example

a(9)=5 and the 5 subsets are {1,2,3,4,5,6,7,8}, {1,2,3,4,5,6,8,9}, {1,2,3,4,6,7,8,9}, {1,2,4,5,6,7,8,9}, {2,3,4,5,6,7,8,9}.

Maple

a:= n-> ((b, q)-> b(q, 4)*b(n-q, 4))(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, 4]*Binomial[(n + 1)/2, 4], Binomial[n/2, 4]^2]; Array[a, 42, 0] (* Amiram Eldar, Jan 21 2020 *)

Prog

(PARI) a(n) = if (n%2, binomial((n-1)/2, 4)*binomial((n+1)/2, 4), binomial(n/2, 4)^2); \\ Michel Marcus, Jan 21 2020
(PARI) concat([0, 0, 0, 0, 0, 0, 0, 0], Vec(x^8*(1 + 3*x + 9*x^2 + 9*x^3 + 9*x^4 + 3*x^5 + x^6) / ((1 - x)^9*(1 + x)^7) + O(x^40))) \\ Colin Barker, Jan 21 2020
(Magma) [IsEven(n) select Binomial((n div 2), 4)^2 else Binomial((n-1) div 2, 4)*Binomial((n+1) div 2, 4): n in [0..41]]; // Marius A. Burtea, Jan 21 2020

Crossrefs

Cf. A028723, A331574.

Cf. A288876 (even bisection, shifted).

Sequence in context: A059302 A147130 A154286 * A078234 A056374 A301912

Adjacent sequences: A331572 A331573 A331574 * A331576 A331577 A331578

Keyword

nonn,easy

Author

Enrique Navarrete, Jan 20 2020

Status

approved