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
KEYWORD
nonn,easy
AUTHOR
Enrique Navarrete, Jan 20 2020
STATUS
approved