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%I #26 Mar 17 2024 11:30:44
%S 1,1,2,3,6,10,20,31,65,101,216,336,728,1128,2472,3823,8451,13051,
%T 29050,44803,100298,154518,347568,534964,1208220,1858156,4211312,
%U 6472168,14712960,22597760,51507280,79067375,180642391,277164295,634551606,973184313,2232223626,3422117190,7862669700,12049586631
%N Number of compositions of n into ceiling(n/2) parts with 1 <= each part <=4.
%F a(n) = sum((-1)^h*binomial(n-4*h-1, ceiling(n/2)-1)*binomial(ceiling(n/2), h), h=0..floor((n-1)/4)).
%F a(n) ~ c * d^(n/2) / sqrt(n), where d = 3.610718613276039349818649008384058627465... is the root of the equation 16 + 8*d + 11 * d^2 - 4*d^3 = 0 and c = sqrt((39 + (4563 - 78*sqrt(78))^(1/3) + (39*(117 + 2*sqrt(78)))^(1/3))/(78*Pi)) = 0.5423866816763379517560447644... if n is even, c = sqrt(24/((-56 + (2*(65228 - 7347*sqrt(78)))^(1/3) + (2*(65228 + 7347*sqrt(78)))^(1/3))*Pi)) = 0.677435919213691192835873220... if n is odd. - _Vaclav Kotesovec_, May 01 2014, updated Mar 17 2024
%e a(4)=3: (1,3),(3,1),(2,2).
%t Tr/@ Table[((-1)^h)*Binomial[n-4h-1,Ceiling[n/2]-1]*Binomial[Ceiling[n/2],h],{n,32},{h,0,Floor[(n-1)/4]}] (* _Wouter Meeussen_, Feb 24 2013 *)
%o (Magma)
%o [&+[(-1)^h*Binomial(n-4*h-1,Ceiling(n/2)-1)*Binomial(Ceiling(n/2),h): h in [0..Floor((n-1)/4)]]: n in [1..40]]; // _Bruno Berselli_, Feb 26 2013
%K nonn
%O 1,3
%A _Shanzhen Gao_, Feb 15 2013
%E a(29) corrected by _Bruno Berselli_, Feb 26 2013
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