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%I #24 Feb 08 2021 17:20:17
%S 1,1,2,4,7,12,22,38,67,118,207,363,638,1119,1964,3447,6049,10615,
%T 18629,32691,57369,100676,176674,310041,544085,954802,1675561,2940405,
%U 5160051,9055258,15890871,27886534,48937456,85879249,150707576,264473359,464118392,814471000,1429296968
%N Expansion of 1/(1-x-x^2-x^3+x^5+x^7).
%C This sequence counts partially ordered partitions of (n) into parts (1,2,3,4) where the position (order) of 3's is unimportant.
%H Robert Israel, <a href="/A257932/b257932.txt">Table of n, a(n) for n = 0..4090</a>
%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,0,-1,0,-1).
%F a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-5) - a(n-7).
%F G.f.: 1 / ((x-1)*(x+1)*(x^2+x+1)*(x^3-x^2+2*x-1)). - _Colin Barker_, May 17 2015
%e a(6)=22; these are (42),(24),(411),(141),(114),(33),(321=231=213),(312=132=123),(3111=1311=1131=1113),(222),(2211),(1122),(1221),(2112),(2121),(1212),(21111),(12111),(11211),(11121),(11112),(111111).
%p f:= gfun:-rectoproc({a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-5) - a(n-7), seq(a(i)=[1, 1, 2, 4, 7, 12,22][i+1],i=0..6)},a(n),remember):
%p map(f, [$0..50]); # _Robert Israel_, Apr 26 2017
%t LinearRecurrence[{1, 1, 1, 0, -1, 0, -1}, {1, 1, 2, 4, 7, 12, 22}, 39] (* _Robert P. P. McKone_, Feb 08 2021 *)
%o (PARI) Vec(1/((x-1)*(x+1)*(x^2+x+1)*(x^3-x^2+2*x-1)) + O(x^100)) \\ _Colin Barker_, May 17 2015
%K nonn,easy
%O 0,3
%A _David Neil McGrath_, May 13 2015
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