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A021724 Expansion of 1/((1-x)(1-3x)(1-10x)(1-12x)).

%I A021724 #25 Jul 08 2025 08:50:29
%S A021724 1,26,465,7150,101621,1378026,18123145,233349350,2958918141,
%T A021724 37094306626,461004657425,5690785933950,69876732453061,
%U A021724 854393804284826,10411455807073305,126524771262956950,1534170271000826381
%N A021724 Expansion of 1/((1-x)(1-3x)(1-10x)(1-12x)).
%C A021724 From _Bruno Berselli_, May 08 2013: (Start)
%C A021724 Naturally, the sequence is related to:
%C A021724 A018207, 1/((1-3x)(1-10x)(1-12x)):  A018207(n) = a(n)-a(n-1), n>0;
%C A021724 A016267, 1/((1-x)(1-10x)(1-12x)):  A016267(n) = a(n)-3*a(n-1), n>0;
%C A021724 A016217, 1/((1-x)(1-3x)(1-12x)):  A016217(n) = a(n)-10*a(n-1), n>0;
%C A021724 A016215, 1/((1-x)(1-3x)(1-10x)):  A016215(n) = a(n)-12*a(n-1), n>0;
%C A021724 A016196, 1/((1-10x)(1-12x)):  A016196(n) = a(n)-4*a(n-1)+3*a(n-2), n>1;
%C A021724 A016147, 1/((1-3x)(1-12x)):  A016147(n) = a(n)-11*a(n-1)+10*a(n-2), n>1;
%C A021724 A016145, 1/((1-3x)(1-10x)):  A016145(n) = a(n)-13*a(n-1)+12*a(n-2), n>1;
%C A021724 A016125, 1/((1-x)(1-12x)):  A016125(n) = a(n)-13*a(n-1)+30*a(n-2), n>1;
%C A021724 A002275, x/((1-x)(1-10x)):  A002275(n) = a(n-1)-15*a(n-2)+36*a(n-3), n>2;
%C A021724 A003462, x/((1-x)(1-3x)):  A003462(n) = a(n-1)-22*a(n-2)+120*a(n-3), n>2;
%C A021724 A000012, 1/(1-x):  A000012(n) = a(n)-25*a(n-1)+186*a(n-2)-360*a(n-3), n>2;
%C A021724 A000244, 1/(1-3x):  A000244(n) = a(n)-23*a(n-1)+142*a(n-2)-120*a(n-3), n>2;
%C A021724 A011557, 1/(1-10x):  A011557(n) = a(n)-16*a(n-1)+51*a(n-2)-36*a(n-3), n>2;
%C A021724 A001021, 1/(1-12x):  A001021(n) = a(n)-14*a(n-1)+43*a(n-2)-30*a(n-3), n>2. (End)
%H A021724 Vincenzo Librandi, <a href="/A021724/b021724.txt">Table of n, a(n) for n = 0..200</a>
%H A021724 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (26,-211,546,-360).
%F A021724 G.f.: 1/((1-x)*(1-3*x)*(1-10*x)*(1-12*x)).
%F A021724 a(n) = -1/198 +3^(n+1)/14 -2^(n+2)*5^(n+3)/63 +2^(2n+5)*3^(n+1)/11. [_Bruno Berselli_, May 07 2013]
%t A021724 CoefficientList[Series[1/((1 - x) (1 - 3 x) (1 - 10 x) (1 - 12 x)), {x, 0, 20}], x] (* _Bruno Berselli_, May 07 2013 *)
%t A021724 LinearRecurrence[{26,-211,546,-360},{1,26,465,7150},120] (* _Harvey P. Dale_, Jul 06 2019 *)
%o A021724 (PARI) Vec(1/((1-x)*(1-3*x)*(1-10*x)*(1-12*x))+O(x^20)) \\ _Bruno Berselli_, May 07 2013
%o A021724 (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)*(1-3*x)*(1-10*x)*(1-12*x)))); // _Bruno Berselli_, May 07 2013
%Y A021724 Cf. A000012, A000244, A001021, A002275, A003462, A011557, A016125, A016145, A016147, A016196, A016215, A016217, A016267, A018207.
%K A021724 nonn,easy
%O A021724 0,2
%A A021724 _N. J. A. Sloane_