A016173 -id:A016173 - cp's OEIS frontend

A016129 Expansion of 1/((1-2x)(1-6*x)).

1, 8, 52, 320, 1936, 11648, 69952, 419840, 2519296, 15116288, 90698752, 544194560, 3265171456, 19591036928, 117546237952, 705277460480, 4231664828416, 25389989101568, 152339934871552, 914039609753600, 5484237659570176, 32905425959518208, 197432555761303552

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (8,-12).

Row sums of A100851.
Cf. A071951, A089278, A089500, A129467, A144843.
Sequences with gf 1/((1-n*x)*(1-6*x)): A000400 (n=0), A003464 (n=1), this sequence (n=2), A016137 (n=3), A016149 (n=4), A005062 (n=5), A053469 (n=6), A016169 (n=7), A016170 (n=8), A016172 (n=9), A016173 (n=10), A016174 (n=11), A016175 (n=12).

[(6^(n+1)-2^(n+1))/4 : n in [0..30]]; // Vincenzo Librandi, Oct 09 2011

Table[(6^(n+1) -2^(n+1))/4, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
CoefficientList[Series[1/((1-2x)(1-6x)),{x,0,30}],x] (* or *) LinearRecurrence[{8,-12},{1,8},30] (* Harvey P. Dale, Jan 15 2015 *)

Vec(1/(1-2*x)/(1-6*x)+O(x^30)) \\ Charles R Greathouse IV, Apr 17 2012

[lucas_number1(n,8,12) for n in range(1, 31)] # Zerinvary Lajos, Apr 23 2009

[(6^n - 2^n)/4 for n in range(1,31)] # Zerinvary Lajos, Jun 04 2009

a(n) = A071951(n+2, 2) = 9*(2*3)^(n-1) - (2*1)^(n-1) = (2^(n-1))*(3^(n+1)-1), n>=0. - Wolfdieter Lang, Nov 07 2003
From Lambert Klasen (lambert.klasen(AT)gmx.net), Feb 05 2005: (Start)
G.f.: 1/((1-2*x)*(1-6*x)).
E.g.f.: (-exp(2*x) + 3*exp(6*x))/2.
a(n) = (6^(n+1) - 2^(n+1))/4. (End)
a(n)^2 = A144843(n+1). - Philippe Deléham, Nov 26 2008
a(n) = 8*a(n-1) - 12*a(n-2). - Philippe Deléham, Jan 01 2009
a(n) = det(|ps(i+2,j+1)|, 1 <= i,j <= n), where ps(n,k) are Legendre-Stirling numbers of the first kind (A129467). - Mircea Merca, Apr 06 2013

A020595 Expansion of g.f. 1/((1-6x)(1-9x)(1-10*x)).

Original entry on oeis.org

1, 25, 421, 5965, 76741, 929005, 10791061, 121699645, 1342777381, 14569879885, 156038219701, 1653799781725, 17380932862021, 181408804717165, 1882561696208341, 19442349988398205, 199976918230722661, 2049766874087336845, 20947749526851028981, 213528831702049245085

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (25,-204,540).

Cf. A016172, A016173.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-6*x)*(1-9*x)*(1-10*x)))); // Vincenzo Librandi, Jul 04 2013

I:=[1, 25, 421]; [n le 3 select I[n] else 25*Self(n-1)-204*Self(n-2)+540*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jul 04 2013

CoefficientList[Series[1 / ((1 - 6 x) (1 - 9 x) (1 - 10 x)), {x, 0, 30}], x] (* or *) LinearRecurrence[{25, -204, 540}, {1, 25, 421}, 20] (* Harvey P. Dale, Oct 13 2012 *)

a(0)=1, a(1)=25, a(2)=421; For n>2, a(n) = 25*a(n-1) - 204*a(n-2) + 540*a(n-3). - Harvey P. Dale, Oct 13 2012
a(n) = (3*10^(n+2) - 4*9^(n+2) + 6^(n+2))/12. - Yahia Kahloune, Jun 30 2013
a(n) = 19*a(n-1) - 90*a(n-2) + 6^n. - Vincenzo Librandi, Jul 04 2013
From Elmo R. Oliveira, Mar 26 2025: (Start)
E.g.f.: exp(6*x)*(300*exp(4*x) - 324*exp(3*x) + 36)/12.
a(n) = A016173(n+1) - A016172(n+1). (End)

A020726 Expansion of g.f. 1/((1-6x)(1-10x)(1-11*x)).

Original entry on oeis.org

1, 27, 493, 7599, 106645, 1411431, 17955757, 222093423, 2690508229, 32080473975, 377794514461, 4405195463487, 50953884924853, 585473143132359, 6690087028209805, 76090252032830991, 861988540696279717, 9731848557669909783, 109550181794434004989, 1230051085699164039135

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (27,-236,660).

Cf. A016173, A016174.

CoefficientList[Series[1/((1-6x)(1-10x)(1-11x)),{x,0,30}],x] (* or *) LinearRecurrence[{27,-236,660},{1,27,493},30] (* Harvey P. Dale, Oct 01 2014 *)

Vec(1/((1-6*x)*(1-10*x)*(1-11*x)) + O(x^30)) \\ Jinyuan Wang, Mar 10 2020

a(n) = 21*a(n-1) - 110*a(n-2) + 6^n for n>1, a(0)=1, a(1)=27. - Vincenzo Librandi, Mar 11 2011
a(n) = (4*11^(n+2) - 5*10^(n+2) + 6^(n+2))/20. - Yahia Kahloune, Jun 30 2013
In general, for the expansion of 1/((1-r*x)(1-s*x)(1-t*x)) with t > s > r, we have the formula: a(n) = ((s-r)*t^(n+2) - (t-r)*s^(n+2) + (t-s)*r^(n+2))/((s-r)*(t-r)*(t-s)). - Yahia Kahloune, Sep 09 2013
a(0) = 1, a(1) = 27, a(2) = 493, a(n) = 27*a(n-1) - 236*a(n-2) + 660*a(n-3). - Harvey P. Dale, Oct 01 2014
From Elmo R. Oliveira, Mar 26 2025: (Start)
E.g.f.: exp(6*x)*(484*exp(5*x) - 500*exp(4*x) + 36)/20.
a(n) = A016174(n+1) - A016173(n+1). (End)

More terms from Elmo R. Oliveira, Mar 26 2025

cp's OEIS Frontend

A016129 Expansion of 1/((1-2x)(1-6*x)).

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A020595 Expansion of g.f. 1/((1-6x)(1-9x)(1-10*x)).

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A020726 Expansion of g.f. 1/((1-6x)(1-10x)(1-11*x)).

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cp's OEIS Frontend

A016129 Expansion of 1/((1-2*x)*(1-6*x)).

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Author

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Magma

Mathematica

PARI

Sage

Sage

Formula

A020595 Expansion of g.f. 1/((1-6*x)*(1-9*x)*(1-10*x)).

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Author

Keywords

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Crossrefs

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Magma

Magma

Mathematica

Formula

A020726 Expansion of g.f. 1/((1-6*x)*(1-10*x)*(1-11*x)).

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Author

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Mathematica

PARI

Formula

Extensions

A016129 Expansion of 1/((1-2x)(1-6*x)).

A020595 Expansion of g.f. 1/((1-6x)(1-9x)(1-10*x)).

A020726 Expansion of g.f. 1/((1-6x)(1-10x)(1-11*x)).