A016174 -id:A016174 - 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

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

A020766 Expansion of g.f. 1/((1-6x)(1-11x)(1-12*x)).

Original entry on oeis.org

1, 29, 571, 9521, 144907, 2083865, 28847827, 388709777, 5134091323, 66784487561, 858403625443, 10928093824193, 138039056180299, 1732402968047417, 21624191213455219, 268679676312195569, 3325242136114316635, 41014868784078912233, 504410121626681853955, 6187470727275006236705

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (29,-270,792).

Cf. A016174, A016175.

CoefficientList[Series[1/((1-6x)(1-11x)(1-12x)),{x,0,20}],x] (* or *) LinearRecurrence[{29,-270,792},{1,29,571},20] (* Harvey P. Dale, Jun 13 2015 *)

a(n) = 23*a(n-1) - 132*a(n-2) + 6^n; a(0)=1, a(1)=29. - Vincenzo Librandi, Mar 11 2011
a(n) = 6*6^n/5 - 121*11^n/5 + 24*12^n. - R. J. Mathar, Jul 01 2013
a(n) = 29*a(n-1) - 270*a(n-2) + 792*a(n-3); a(0)=1, a(1)=29, a(2)=571. - Harvey P. Dale, Jun 13 2015
From Elmo R. Oliveira, Mar 26 2025: (Start)
E.g.f.: exp(6*x)*(6 - 121*exp(5*x) + 120*exp(6*x))/5.
a(n) = A016175(n+1) - A016174(n+1). (End)

More terms from Elmo R. Oliveira, Mar 26 2025

A102765 Array read by antidiagonals: T(n, k) = ((n+4)^k-(n-1)^k)/5.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 5, 13, 0, 1, 7, 25, 51, 0, 1, 9, 43, 125, 205, 0, 1, 11, 67, 259, 625, 819, 0, 1, 13, 97, 477, 1555, 3125, 3277, 0, 1, 15, 133, 803, 3355, 9331, 15625, 13107, 0, 1, 17, 175, 1261, 6505, 23517, 55987, 78125, 52429, 0, 1, 19, 223, 1875, 11605

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Feb 10 2005

Consider a 5x5 matrix M =
[n, 1, 1, 1, 1]
[1, n, 1, 1, 1]
[1, 1, n, 1, 1]
[1, 1, 1, n, 1]
[1, 1, 1, 1, n].
The n-th row of the array contains the values of the non diagonal elements of M^k, k=0,1,.... (Corresponding diagonal entry = non diagonal entry + (n-1)^k.)
For row r we have polynomial ((r+4)^n-(r-1)^n)/5. Corresponding g.f.s: x/((1-(r-1)x)(1-(r+4)x))
If r(n) denotes a row sequence, r(n+1)/r(n) converges to n+4.
Triangle T(n, k) = (4^(n-k-1)-(-1)^(n-k-1))/5*(binomial(k+(n-k-1),n-k-1)) gives coefficients for polynomials for the columns of the array. First four polynomial are:
  1
  3 + 2*k
  13 + 9*k + 3*k^2
  51 + 52*k + 18*k^2 + 4*k^3
  ...

			Array begins:
  0, 1,  3, 13,  51,  205, ...
  0, 1,  5, 25, 125,  625, ...
  0, 1,  7, 43, 259, 1555, ...
  0, 1,  9, 67, 477, 3355, ...
  0, 1, 11, 97, 803, 6505, ...
  ...

Cf. A015521 (for n=0), A000351 (for n=1), A003464 (for n=2), A016130 (for n=3), A016140 (for n=4), A016153 (for n=5), A016164 (for n=6), A016174 (for n=7), A016184 (for n=8), A015441 (for n=-1), A091005 (for n=-2).

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(5,k)^i)[1,2],","));print())

p(n,k)=((n+4)^k-(n-1)^k)/5
for(k=0,10, for(i=0,10,print1(p(k,i),","));print())

for(k=0,10, for(i=0,10,print1(polcoeff(x/((1-(k-1)*x)*(1-(k+4)*x)),i),","));print())

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

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

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A020766 Expansion of g.f. 1/((1-6*x)*(1-11*x)*(1-12*x)).

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A102765 Array read by antidiagonals: T(n, k) = ((n+4)^k-(n-1)^k)/5.

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

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

A020766 Expansion of g.f. 1/((1-6x)(1-11x)(1-12*x)).