A016172 -id:A016172 - 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

A020579 Expansion of g.f. 1/((1-6x)(1-8x)(1-9*x)).

Original entry on oeis.org

1, 23, 355, 4595, 53851, 592403, 6240235, 63710915, 635468251, 6225852083, 60146237515, 574587484835, 5439634923451, 51116555484563, 477406092913195, 4435981769620355, 41041272503703451, 378327871809737843, 3476703760455563275, 31864966517183461475, 291385416197758352251

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 (23,-174,432).

Cf. A016170, A016172.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-6*x)*(1-8*x)*(1-9*x)))); // Vincenzo Librandi, Jul 04 2013

I:=[1,23,355]; [n le 3 select I[n] else 23*Self(n-1)-174*Self(n-2)+432*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jul 04 2013

CoefficientList[Series[1 / ((1 - 6 x) (1 - 8 x) (1 - 9 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jul 04 2013 *)

a(n) = 6*6^n - 32*8^n + 27*9^n. - R. J. Mathar, Jun 30 2013
From Vincenzo Librandi, Jul 04 2013: (Start)
a(0)=1, a(1)=23, a(2)=355; for n>2, a(n) = 23*a(n-1) - 174*a(n-2) + 432*a(n-3).
a(n) = 17*a(n-1) - 72*a(n-2) + 6^n. (End)
From Elmo R. Oliveira, Mar 26 2025: (Start)
E.g.f.: exp(6*x)*(6 - 32*exp(2*x) + 27*exp(3*x)).
a(n) = A016172(n+1) - A016170(n+1). (End)

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)

A102752 Array read by antidiagonals: T(n, k) = ((n+2)^k-(n-1)^k)/3.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 3, 3, 0, 1, 5, 9, 5, 0, 1, 7, 21, 27, 11, 0, 1, 9, 39, 85, 81, 21, 0, 1, 11, 63, 203, 341, 243, 43, 0, 1, 13, 93, 405, 1031, 1365, 729, 85, 0, 1, 15, 129, 715, 2511, 5187, 5461, 2187, 171, 0, 1, 17, 171, 1157, 5261, 15309, 25999, 21845, 6561, 341, 0, 1

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Feb 09 2005

Consider a 3 X 3 matrix M =
  [n, 1, 1]
  [1, n, 1]
  [1, 1, n].
The n-th row of the array contains the values of the nondiagonal elements of M^k, k=0,1,.... (Corresponding diagonal entry = nondiagonal entry + (n-1)^k.)
Table:
  n: row sequence G.f. cross references.
  0: (2^n-(-1)^n)/3 1/((1+1x)(1-2x)) A001045 (Jacobsthal sequence)
  1: (3^n-0^n)/3 1/(1-3x) A000244
  2: (4^n-1^n)/3 1/((1-1x)(1-4x)) A002450
  3: (5^n-2^n)/3 1/((1-2x)(1-5x)) A016127
  4: (6^n-3^n)/3 1/((1-3x)(1-6x)) A016137
  5: (7^n-4^n)/3 1/((1-4x)(1-7x)) A016150
  6: (8^n-5^n)/3 1/((1-5x)(1-8x)) A016162
  7: (9^n-6^n)/3 1/((1-6x)(1-9x)) A016172
  8: (10^n-7^n)/3 1/((1-7x)(1-10x)) A016181
  9: (11^n-8^n)/3 1/((1-8x)(1-11x)) A016187
  10:(12^n-9^n)/3 1/((1-9x)(1-12x)) A016191
If r(n) denotes a row sequence, r(n+1)/r(n) converges to n+2.
Columns follow polynomials with certain binomial coefficients:
  column: polynomial
  0: 0
  1: 1
  2: 2*n + 1
  3: 3*n^2+ 3*n + 3
  4: 4*n^3+ 6*n^2+ 12*n + 5
  5: 5*n^4+10*n^3+ 30*n^2+ 25*n + 11
  6: 6*n^5+15*n^4+ 60*n^3+ 75*n^2+ 66*n + 21
  7: 7*n^6+21*n^5+105*n^4+ 175*n^3+ 231*n^2+ 147*n + 43
  8: 8*n^7+28*n^6+168*n^5+ 350*n^4+ 616*n^3+ 588*n^2+344*n+ 85
  etc.
Coefficients are generated by the array T(n,k)=(2^(n-k-1)-(-1)^(n-k-1))/3*(binomial(k+(n-k-1),n-k-1)) read by antidiagonals.

			Array begins:
  0, 1, 1,  3,   5,   11, ...
  0, 1, 3,  9,  27,   81, ...
  0, 1, 5, 21,  85,  341, ...
  0, 1, 7, 39, 203, 1031, ...
  0, 1, 9, 63, 405, 2511, ...
  ...

MM(n,N)=local(M);M=matrix(n,n);for(i=1,n, for(j=1,n,if(i==j,M[i,j]=N,M[i,j]=1)));M for(k=0,10, for(i=0,10,print1((MM(3,k)^i)[1,2],","));print())

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

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A020579 Expansion of g.f. 1/((1-6*x)*(1-8*x)*(1-9*x)).

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

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A102752 Array read by antidiagonals: T(n, k) = ((n+2)^k-(n-1)^k)/3.

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

A020579 Expansion of g.f. 1/((1-6x)(1-8x)(1-9*x)).

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