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A029032 Expansion of 1/((1-x)*(1-x^3)*(1-x^4)*(1-x^5)).

%I A029032 #35 Jul 08 2025 19:01:27
%S A029032 1,1,1,2,3,4,5,6,8,10,12,14,17,20,23,27,31,35,40,45,51,57,63,70,78,86,
%T A029032 94,103,113,123,134,145,157,170,183,197,212,227,243,260,278,296,315,
%U A029032 335,356,378,400,423,448,473,499,526,554,583,613,644,676,709,743
%N A029032 Expansion of 1/((1-x)*(1-x^3)*(1-x^4)*(1-x^5)).
%C A029032 a(n) is the number of partitions of n into parts 1, 3, 4, and 5. - _David Neil McGrath_, Sep 13 2014
%H A029032 Hoang Xuan Thanh, <a href="/A029032/b029032.txt">Table of n, a(n) for n = 0..10000</a>
%H A029032 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,0,0,-1,-1,0,0,1,0,1,-1).
%F A029032 a(0)=1, a(1)=1, a(2)=1, a(3)=2, a(4)=3, a(5)=4, a(6)=5, a(7)=6, a(8)=8, a(9)=10, a(10)=12, a(11)=14, a(12)=17, a(n)=a(n-1)+a(n-3)-a(n-6)- a(n-7)+ a(n-10)+a(n-12)-a (n-13). - _Harvey P. Dale_, Jan 04 2012
%F A029032 From _R. J. Mathar_, Jun 23 2021: (Start)
%F A029032 a(n)-a(n-1) = A008680(n).
%F A029032 a(n)-a(n-3) = A025772(n).
%F A029032 a(n)-a(n-4) = A008672(n).
%F A029032 a(n)-a(n-5) = A025767(n). (End)
%F A029032 a(n) = 1 + floor((2*n^3+39*n^2+228*n)/720). - _Hoang Xuan Thanh_, May 29 2025
%p A029032 M := Matrix(13, (i,j)-> if (i=j-1) or (j=1 and member(i, [1, 3, 10, 12])) then 1 elif j=1 and member(i, [6, 7, 13]) then -1 else 0 fi); a := n -> (M^(n))[1,1]; seq (a(n), n=0..49); # _Alois P. Heinz_, Jul 25 2008
%t A029032 CoefficientList[Series[1/((1-x)(1-x^3)(1-x^4)(1-x^5)),{x,0,50}],x] (* _Harvey P. Dale_, Jan 04 2012 *)
%K A029032 nonn,easy
%O A029032 0,4
%A A029032 _N. J. A. Sloane_