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%I #20 Apr 08 2024 06:56:29
%S 1,64,1094,8192,39063,139968,411772,1048576,2391485,5000000,9743586,
%T 17915904,31374259,52706752,85429688,134217728,205169337,306110016,
%U 446935870,640000000,900544271,1247178944,1702412724,2293235712,3051757813,4015905088
%N Number of compositions of even natural numbers in 7 parts <= n.
%C Number of ways of placing an even number of indistinguishable objects in 7 distinguishable boxes with the condition that in each box can be at most n objects.
%H Vincenzo Librandi, <a href="/A191494/b191494.txt">Table of n, a(n) for n = 0..1000</a>
%H Adi Dani, <a href="https://oeis.org/wiki/User:Adi_Dani_/Restricted_compositions_of_natural_numbers">Restricted compositions of natural numbers</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (7,-20,28,-14,-14,28,-20,7,-1).
%F a(n) = ((n + 1)^7 + (1 + (-1)^n)/2 )/2.
%F G.f.: ( 1 + 57*x + 666*x^2 + 1786*x^3 + 1821*x^4 + 645*x^5 + 64*x^6 ) / ( (1+x)*(x-1)^8 ). - _R. J. Mathar_, Jun 08 2011
%e a(1)=64 and compositions of even natural numbers into 7 parts no greater than 1 are
%e :(0,0,0,0,0,0,0) --> 7!/7!0! = 1
%e :(0,0,0,0,0,1,1) --> 7!/5!2! = 21
%e :(0,0,0,1,1,1,1) --> 7!/3!4! = 35
%e :(0,1,1,1,1,1,1) --> 7!/1!6! = 7
%t Table[1/2*((n + 1)^7 + (1 + (-1)^n)*1/2), {n, 0, 25}]
%o (Magma) [((n + 1)^7 + (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), A191489 (6 parts), A191495 (8 parts).
%K nonn,easy,changed
%O 0,2
%A _Adi Dani_, Jun 03 2011
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