A191484 Number of compositions of even natural numbers into 5 parts <= n.

1, 16, 122, 512, 1563, 3888, 8404, 16384, 29525, 50000, 80526, 124416, 185647, 268912, 379688, 524288, 709929, 944784, 1238050, 1600000, 2042051, 2576816, 3218172, 3981312, 4882813, 5940688

Offset: 0

Author

Adi Dani, Jun 03 2011

Keywords

• nonn

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.

Examples

			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).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Adi Dani, Restricted compositions of natural numbers
• Index entries for linear recurrences with constant coefficients, signature (5,-9,5,5,-9,5,-1).

Crossrefs

Cf. A036486 (3 parts), A171714 (4 parts), A191489 (6 parts), A191494 (7 parts), A191495 (8 parts).

Programs

Magma

[((n + 1)^5 + (1 + (-1)^n)/2 )/2: n in [0..40]]; // Vincenzo Librandi, Jun 16 2011

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 *)

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)).