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

%I #46 Sep 08 2022 08:44:36

%S 1,0,0,1,1,1,1,1,2,2,2,2,3,3,3,4,4,4,5,5,6,6,6,7,8,8,8,9,10,10,11,11,

%T 12,13,13,14,15,15,16,17,18,18,19,20,21,22,22,23,25,25,26,27,28,29,30,

%U 31,32,33,34,35,37,37,38,40,41,42,43,44,46,47,48,49,51,52,53,55,56,57

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

%C Number of partitions of n into parts 3, 4, and 5. - _Joerg Arndt_, Aug 17 2013

%H Seiichi Manyama, <a href="/A008680/b008680.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Vincenzo Librandi)

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=228">Encyclopedia of Combinatorial Structures 228</a>

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1,1,1,0,-1,-1,-1,0,0,1).

%F Euler transform of length 5 sequence [ 0, 0, 1, 1, 1]. - _Michael Somos_, Aug 13 2007

%F From _Michael Somos_, Aug 13 2007: (Start)

%F G.f.: 1 / ((1 - x^3) * (1 - x^4) * (1 - x^5)).

%F a(n) = a(-12-n) for all n in Z. (End)

%F a(n) = floor((1+(-1)^n)*(-1)^floor(n/2)/8 +(n^2+12*n+90)/120). - _Tani Akinari_, Aug 17 2013

%e G.f. = 1 + x^3 + x^4 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + 2*x^11 + ...

%p a:= proc(n) local m, r; m:= iquo(n, 60, 'r'); r:= r+1; (5+r+30*m)*m+ [1, 0$2, 1$5, 2$4, 3$3, 4$3, 5$2, 6$3, 7, 8$3, 9, 10$2, 11$2, 12, 13$2, 14, 15$2, 16, 17, 18$2, 19, 20, 21, 22$2, 23, 25, i$i=25..35][r] end: seq(a(n), n=0..100); # _Alois P. Heinz_, Oct 06 2008

%t CoefficientList[Series[1/((1-x^3)(1-x^4)(1-x^5)),{x,0,80}],x] (* _Harvey P. Dale_, Apr 29 2011 *)

%o (PARI) {a(n) = if( n<0, n=-12-n); polcoeff( 1 / ((1 - x^3) * (1 - x^4) * (1 - x^5)) + x * O(x^n), n)}; /* _Michael Somos_, Aug 13 2007 */

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x^3)*(1-x^4)*(1-x^5)) )); // _G. C. Greubel_, Sep 09 2019

%o (Sage)

%o def A008680_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P(1/((1-x^3)*(1-x^4)*(1-x^5))).list()

%o A008680_list(80) # _G. C. Greubel_, Sep 09 2019

%o (GAP) a:=[1,0,0,1,1,1,1,1,2,2,2,2];; for n in [13..80] do a[n]:=a[n-3] +a[n-4]+a[n-5]-a[n-7]-a[n-8]-a[n-9]+a[n-12]; od; a; # _G. C. Greubel_, Sep 09 2019

%K nonn,easy

%O 0,9

%A _N. J. A. Sloane_

%E Typo in name fixed by _Vincenzo Librandi_, Jun 23 2013