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Number of compositions (ordered partitions) of n into 4 parts, where the first is at least as great as each of the others.
5

%I #30 May 17 2018 13:43:27

%S 1,1,4,7,11,17,26,35,48,63,81,102,127,154,187,223,263,308,359,413,474,

%T 540,612,690,775,865,964,1069,1181,1301,1430,1565,1710,1863,2025,2196,

%U 2377,2566,2767,2977,3197,3428,3671,3923,4188,4464,4752,5052,5365,5689

%N Number of compositions (ordered partitions) of n into 4 parts, where the first is at least as great as each of the others.

%C For n=1,2 these are just the tetrahedral numbers. a(n) is always at least 1/4 of the corresponding tetrahedral number, since each partition of this type gives up to four ordered partitions with the same cyclical order.

%C Diagonal sums of the irregular triangle A109439, for example a(0)=1, a(1)=1, a(2)=1+3, a(3)=1+3+3, a(4)=1+3+6+1. - _Bob Selcoe_, Feb 09 2014

%H Alois P. Heinz, <a href="/A156039/b156039.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (2, -1, 1, -1, -1, 1, -1, 2, -1).

%F G.f.: ( 1-x+3*x^2-x^3+x^4 ) / ( (1+x)*(1+x^2)*(1+x+x^2)*(x-1)^4 ). - _Alois P. Heinz_, Jun 14 2009

%e For n = 3 the a(3) = 7 compositions are: (3 0 0 0) (2 1 0 0) (2 0 1 0) (2 0 0 1) (1 1 1 0) (1 1 0 1) (1 0 1 1).

%p a:= proc(n) local m, r; m:= iquo(n, 12, 'r'); r:= r+1; (9 +(27 +72*m +18*r)*m +((9 +3*r) *r-12) /2)*m +[1, 1, 4, 7, 11, 17, 26, 35, 48, 63, 81, 102][r] end: seq(a(n), n=0..60); # _Alois P. Heinz_, Jun 14 2009

%t LinearRecurrence[{2, -1, 1, -1, -1, 1, -1, 2, -1}, {1, 1, 4, 7, 11, 17, 26, 35, 48}, 60] (* _Jean-François Alcover_, May 17 2018 *)

%Y For partitions into 3 summands see A156040; also see A156041 and A156042.

%K nonn

%O 0,3

%A _Jack W Grahl_, Feb 02 2009, Feb 11 2009

%E More terms from _Alois P. Heinz_, Jun 14 2009