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

1, 26, 465, 7150, 101621, 1378026, 18123145, 233349350, 2958918141, 37094306626, 461004657425, 5690785933950, 69876732453061, 854393804284826, 10411455807073305, 126524771262956950, 1534170271000826381

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

From Bruno Berselli, May 08 2013: (Start)

Naturally, the sequence is related to:

A018207, 1/((1-3x)(1-10x)(1-12x)): A018207(n) = a(n)-a(n-1), n>0;

A016267, 1/((1-x)(1-10x)(1-12x)): A016267(n) = a(n)-3*a(n-1), n>0;

A016217, 1/((1-x)(1-3x)(1-12x)): A016217(n) = a(n)-10*a(n-1), n>0;

A016215, 1/((1-x)(1-3x)(1-10x)): A016215(n) = a(n)-12*a(n-1), n>0;

A016196, 1/((1-10x)(1-12x)): A016196(n) = a(n)-4*a(n-1)+3*a(n-2), n>1;

A016147, 1/((1-3x)(1-12x)): A016147(n) = a(n)-11*a(n-1)+10*a(n-2), n>1;

A016145, 1/((1-3x)(1-10x)): A016145(n) = a(n)-13*a(n-1)+12*a(n-2), n>1;

A016125, 1/((1-x)(1-12x)): A016125(n) = a(n)-13*a(n-1)+30*a(n-2), n>1;

A002275, x/((1-x)(1-10x)): A002275(n) = a(n-1)-15*a(n-2)+36*a(n-3), n>2;

A003462, x/((1-x)(1-3x)): A003462(n) = a(n-1)-22*a(n-2)+120*a(n-3), n>2;

A000012, 1/(1-x): A000012(n) = a(n)-25*a(n-1)+186*a(n-2)-360*a(n-3), n>2;

A000244, 1/(1-3x): A000244(n) = a(n)-23*a(n-1)+142*a(n-2)-120*a(n-3), n>2;

A011557, 1/(1-10x): A011557(n) = a(n)-16*a(n-1)+51*a(n-2)-36*a(n-3), n>2;

A001021, 1/(1-12x): A001021(n) = a(n)-14*a(n-1)+43*a(n-2)-30*a(n-3), n>2. (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (26,-211,546,-360).

Crossrefs

Cf. A000012, A000244, A001021, A002275, A003462, A011557, A016125, A016145, A016147, A016196, A016215, A016217, A016267, A018207.

Programs

Magma

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)*(1-3*x)*(1-10*x)*(1-12*x)))); // Bruno Berselli, May 07 2013

Mathematica

CoefficientList[Series[1/((1 - x) (1 - 3 x) (1 - 10 x) (1 - 12 x)), {x, 0, 20}], x] (* Bruno Berselli, May 07 2013 *)
LinearRecurrence[{26,-211,546,-360},{1,26,465,7150},120] (* Harvey P. Dale, Jul 06 2019 *)

PARI

Vec(1/((1-x)*(1-3*x)*(1-10*x)*(1-12*x))+O(x^20)) \\ Bruno Berselli, May 07 2013

Formula

G.f.: 1/((1-x)*(1-3*x)*(1-10*x)*(1-12*x)).

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]