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

%I A191484 #26 Apr 08 2024 06:56:56
%S A191484 1,16,122,512,1563,3888,8404,16384,29525,50000,80526,124416,185647,
%T A191484 268912,379688,524288,709929,944784,1238050,1600000,2042051,2576816,
%U A191484 3218172,3981312,4882813,5940688
%N A191484 Number of compositions of even natural numbers into 5 parts <= n.
%C A191484 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.
%H A191484 Vincenzo Librandi, <a href="/A191484/b191484.txt">Table of n, a(n) for n = 0..1000</a>
%H A191484 Adi Dani, <a href="https://oeis.org/wiki/User:Adi_Dani_/Restricted_compositions_of_natural_numbers">Restricted compositions of natural numbers</a>
%H A191484 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,5,5,-9,5,-1).
%F A191484 a(n) = ((n + 1)^5 + (1 + (-1)^n)/2 )/2.
%F A191484 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).
%F A191484 G.f.: (16*x^4 + 41*x^3 + 51*x^2 + 11*x + 1)/((1-x)^6*(1+x)).
%e A191484 a(1)=16 as there are 16 compositions of even natural numbers into 5 parts <= 1:
%e A191484 (0,0,0,0,0);
%e A191484 (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);
%e A191484 (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).
%t A191484 Table[1/2*((n + 1)^5 + (1 + (-1)^n)*1/2), {n, 0, 25}]
%t A191484 LinearRecurrence[{5,-9,5,5,-9,5,-1},{1,16,122,512,1563,3888,8404},50] (* _Harvey P. Dale_, Nov 09 2011 *)
%o A191484 (Magma) [((n + 1)^5 + (1 + (-1)^n)/2 )/2: n in [0..40]]; // _Vincenzo Librandi_, Jun 16 2011
%Y A191484 Cf. A036486 (3 parts), A171714 (4 parts), A191489 (6 parts), A191494 (7 parts), A191495 (8 parts).
%K A191484 nonn
%O A191484 0,2
%A A191484 _Adi Dani_, Jun 03 2011