A214002 Number of compositions of n into ceiling(n/2) parts with 1 <= each part <=4.

1, 1, 2, 3, 6, 10, 20, 31, 65, 101, 216, 336, 728, 1128, 2472, 3823, 8451, 13051, 29050, 44803, 100298, 154518, 347568, 534964, 1208220, 1858156, 4211312, 6472168, 14712960, 22597760, 51507280, 79067375, 180642391, 277164295, 634551606, 973184313, 2232223626, 3422117190, 7862669700, 12049586631

Offset

1,3

Links

Table of n, a(n) for n=1..40.

Formula

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)).

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

Example

a(4)=3: (1,3),(3,1),(2,2).

Mathematica

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 *)

Prog

(Magma)
[&+[(-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

Crossrefs

Sequence in context: A342012 A050291 A324739 * A305889 A135452 A077027

Adjacent sequences: A213999 A214000 A214001 * A214003 A214004 A214005

Keyword

nonn

Author

Shanzhen Gao, Feb 15 2013

Extensions

a(29) corrected by Bruno Berselli, Feb 26 2013

Status

approved