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%I #24 Apr 08 2024 06:56:48
%S 1,32,365,2048,7813,23328,58825,131072,265721,500000,885781,1492992,
%T 2413405,3764768,5695313,8388608,12068785,17006112,23522941,32000000,
%U 42883061,56689952,74017945,95551488,122070313,154457888
%N Number of compositions of even natural numbers into 6 parts <= n.
%C Number of ways of placing of an even number of indistinguishable objects in 6 distinguishable boxes with condition that in each box can be at most n objects.
%H Vincenzo Librandi, <a href="/A191489/b191489.txt">Table of n, a(n) for n = 0..1000</a>
%H Adi Dani, <a href="http://oeis.org/wiki/User:Adi_Dani">Restricted compositions of natural numbers</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (6,-14,14,0,-14,14,-6,1).
%F a(n) = ((n + 1)^6 + (1+(-1)^n)/2 )/2.
%F G.f.: (x^2 + 10*x + 1)*(x^4 + 16*x^3 + 26*x^2 + 16*x + 1) / ( (1+x)*(1-x)^7 ). - _R. J. Mathar_, Jun 06 2011
%F a(2n) = A175113(n). - _R. J. Mathar_, Jun 07 2011
%e a(1)=32 compositions of even natural numbers in 6 parts <= 1 are
%e :(0,0,0,0,0,0)--> 6!/(6!0!) = 1
%e :(0,0,0,0,1,1)--> 6!/(4!2!) = 15
%e :(0,0,1,1,1,1)--> 6!/(2!4!) = 15
%e :(1,1,1,1,1,1)--> 6!/(0!6!) = 1
%e a(2)=365 compositions of even natural numbers in 6 parts <= 2 are
%e :(0,0,0,0,0,0)--> 6!/(6!0!0!) = 1
%e :(0,0,0,0,1,1)--> 6!/(4!2!0!) = 15
%e :(0,0,0,0,0,2)--> 6!/(5!0!1!) = 6
%e :(0,0,1,1,1,1)--> 6!/(2!4!0!) = 15
%e :(0,0,0,1,1,2)--> 6!/(3!2!1!) = 60
%e :(0,0,0,0,2,2)--> 6!/(4!0!2!) = 15
%e :(0,1,1,1,1,2)--> 6!/(1!4!1!) = 30
%e :(0,0,0,2,2,2)--> 6!/(3!0!3!) = 20
%e :(0,0,1,1,2,2)--> 6!/(2!2!2!) = 90
%e :(1,1,1,1,1,1)--> 6!/(0!6!0!) = 1
%e :(0,1,1,2,2,2)--> 6!/(1!2!3!) = 60
%e :(0,0,2,2,2,2)--> 6!/(2!0!4!) = 15
%e :(1,1,1,1,2,2)--> 6!/(0!4!2!) = 15
%e :(0,2,2,2,2,2)--> 6!/(1!0!5!) = 6
%e :(1,1,2,2,2,2)--> 6!/(0!2!4!) = 15
%e :(2,2,2,2,2,2)--> 6!/(0!0!6!) = 1
%t Table[1/2*((n + 1)^6 + (1 + (-1)^n)*1/2), {n, 0, 25}]
%o (Magma) [((n + 1)^6 + (1+(-1)^n)/2 )/2: n in [0..40]]; // _Vincenzo Librandi_, Jun 16 2011
%Y Cf. A036486 (3 parts), A171714 (4 parts), A191484 (5 parts), A191494 (7 parts), A191495 (8 parts).
%K nonn,easy
%O 0,2
%A _Adi Dani_, Jun 03 2011
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