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%I #29 Sep 08 2022 08:44:50
%S 1,1,1,1,1,1,1,1,2,3,3,3,4,4,4,4,5,6,7,7,8,9,9,9,11,12,13,14,15,16,17,
%T 17,19,21,22,23,26,27,28,29,31,33,35,36,39,42,43,44,48,50,52,54,57,60,
%U 63,64,68,72,74,76,81,84
%N Expansion of 1/((1-x)*(1-x^8)*(1-x^9)*(1-x^12)).
%C Number of partitions of n into parts 1, 8, 9, and 12. - _Joerg Arndt_, Jul 04 2014
%H Robert Israel, <a href="/A029129/b029129.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_30">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, -1, 1, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 1, -1).
%p # this requires Maple 17 or later
%p N:= 10000: # to get a(0) to a(N)
%p with(SignalProcessing):
%p A1:= Array(0..N,1):
%p A8:=Array(0..N, i -> `if`(i mod 8 = 0,1,0)):
%p A9:=Array(0..N, i -> `if`(i mod 9 = 0,1,0)):
%p A12:= Array(0..N,i -> `if`(i mod 12 = 0,1,0)):
%p B:= Convolution(A1,A8)[1..N+1]:
%p B:= Convolution(B, A9)[1..N+1]:
%p B:= Convolution(B, A12)[1..N+1]:
%p A:= map(round,ArrayTools:-Reshape(B,0..N)):
%p seq(A[n],n=0..N); # _Robert Israel_, Jun 03 2014
%t CoefficientList[Series[1/((1-x)*(1-x^8)*(1-x^9)*(1-x^12)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jul 04 2014 *)
%o (Maxima) a(n):=floor((2*n^3+90*n^2+21*n+13023)/10368+(n+15)*(-1)^n/384 +((n+10)*(1-(-1)^n)/2+5)*((-1)^floor(n/2))/96 +(floor(n/3) -floor((n-1)/3))*(n+3)/108 +(floor((n+2)/9) +floor((n+1)/9) +floor(n/9))/3)$ makelist(a(n),n,0,10000); /* _Tani Akinari_, Jul 03 2014 */
%o (Magma) m:=70; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)*(1-x^8)*(1-x^9)*(1-x^12)))); // _Bruno Berselli_, Jul 04 2014
%o (Sage) m = 70; L.<x> = PowerSeriesRing(ZZ, m); f = 1/((1-x)*(1-x^8)*(1-x^9)*(1-x^12)); print(f.coefficients()) # _Bruno Berselli_, Jul 04 2014
%o (PARI) Vec(1/((1-x)*(1-x^8)*(1-x^9)*(1-x^12)) + O(x^70)) \\ _Michel Marcus_, Jul 04 2014
%K nonn,easy
%O 0,9
%A _N. J. A. Sloane_, Dec 11 1999