OFFSET
0,2
COMMENTS
Number of ways of placing an even number of indistinguishable objects in 5 distinguishable boxes with the condition that in each box can be at most n objects.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-9,5,5,-9,5,-1).
FORMULA
a(n) = ((n + 1)^5 + (1 + (-1)^n)/2 )/2.
a(n) = 5*a(n-1) - 9*a(n-2) + 5*a(n-3) + 5*a(n-4) - 9*a(n-5) + 5*a(n-6) - a(n-7).
G.f.: (16*x^4 + 41*x^3 + 51*x^2 + 11*x + 1)/((1-x)^6*(1+x)).
EXAMPLE
a(1)=16 as there are 16 compositions of even natural numbers into 5 parts <= 1:
(0,0,0,0,0);
(0,0,0,1,1), (0,0,1,0,1), (0,0,1,1,0), (0,1,1,0,0), (0,1,0,1,0), (0,1,0,0,1), (1,1,0,0,0), (1,0,1,0,0), (1,0,0,1,0), (1,0,0,0,1);
(0,1,1,1,1), (1,0,1,1,1), (1,1,0,1,1), (1,1,1,0,1), (1,1,1,1,0).
MATHEMATICA
Table[1/2*((n + 1)^5 + (1 + (-1)^n)*1/2), {n, 0, 25}]
LinearRecurrence[{5, -9, 5, 5, -9, 5, -1}, {1, 16, 122, 512, 1563, 3888, 8404}, 50] (* Harvey P. Dale, Nov 09 2011 *)
PROG
(Magma) [((n + 1)^5 + (1 + (-1)^n)/2 )/2: n in [0..40]]; // Vincenzo Librandi, Jun 16 2011
CROSSREFS
KEYWORD
nonn
AUTHOR
Adi Dani, Jun 03 2011
STATUS
approved