A016137 -id:A016137 - 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

A075498 Stirling2 triangle with scaled diagonals (powers of 3).

Original entry on oeis.org

1, 3, 1, 9, 9, 1, 27, 63, 18, 1, 81, 405, 225, 30, 1, 243, 2511, 2430, 585, 45, 1, 729, 15309, 24381, 9450, 1260, 63, 1, 2187, 92583, 234738, 137781, 28350, 2394, 84, 1, 6561, 557685, 2205225, 1888110, 563031, 71442, 4158, 108, 1

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the D. E. Knuth reference given in A039692 for exponential convolution arrays.
The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(3*z) - 1)*x/3) - 1.
Subtriangle of the triangle given by (0, 3, 0, 6, 0, 9, 0, 12, 0, 15, 0, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938, see example. - Philippe Deléham, Feb 13 2013
Also the Bell transform of A000244. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

			[1]; [3,1]; [9,9,1]; ...; p(3,x) = x*(9 + 9*x + x^2).
From _Philippe Deléham_, Feb 13 2013: (Start)
Triangle (0, 3, 0, 6, 0, 9, 0, 12, 0, 15, 0, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, ...) begins:
  1;
  0,   1;
  0,   3,   1;
  0,   9,   9,   1;
  0,  27,  63,  18,   1;
  0,  81, 405, 225,  30,   1;
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 1-7 are A000244, A016137, A017933, A028085, A075515, A075516, A075906. Row sums are A004212.

Cf. A075497, A075499.

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> 3^n, 9); # Peter Luschny, Jan 26 2016

Flatten[Table[3^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)
rows = 9;
t = Table[3^n, {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

for(n=1, 11, for(m=1, n, print1(3^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

a(n, m) = (3^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*3)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 3*m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-3*k*x), m >= 1.
E.g.f. for m-th column: (((exp(3*x)-1)/3)^m)/m!, m >= 1.
From Peter Bala, Jan 13 2018: (Start)
Dobinski-type formulas for row polynomials R(n,x):
R(n,x) = exp(-x/3)*Sum_{i >= 0} (3*i)^n* (x/3)^i/i!;
R(n+1,x) = x*exp(-x/3)*Sum_{i >= 0} (3 + 3*i)^n* (x/3)^i/i!.
R(n+1,x) = x*Sum_{k = 0..n} binomial(n,k)*3^(n-k)*R(k,x).(End)

A017933 Expansion of 1/((1-3x)(1-6x)(1-9x)).

Original entry on oeis.org

1, 18, 225, 2430, 24381, 234738, 2205225, 20404710, 186995061, 1703091258, 15448694625, 139763668590, 1262226050541, 11386154248578, 102632111782425, 924629361662070, 8327306431726821, 74979611075290698

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 (18,-99,162).

Third column of triangle A075498.

Cf. A016137, A028085.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-6*x)*(1-9*x)))); /* or */ I:=[1, 18, 225]; [n le 3 select I[n] else 18*Self(n-1)-99*Self(n-2)+162*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jul 02 2013

CoefficientList[Series[1 / ((1 - 3 x) (1 - 6 x) (1 - 9 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Jul 02 2013 *)
LinearRecurrence[{18,-99,162},{1,18,225},20] (* Harvey P. Dale, Sep 09 2023 *)

Vec(1/((1-3*x)*(1-6*x)*(1-9*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = (3^n)*Stirling2(n+3, 3), n >= 0, with Stirling2(n, m) = A008277(n, m).
a(n) = (3^n - 8*6^n + 9*9^n)/2.
G.f.: 1/((1-3*x)*(1-6*x)*(1-9*x)).
E.g.f.: (d^3/dx^3)((((exp(3*x)-1)/3)^3)/3!) = (exp(3*x) - 8*exp(6*x) + 9*exp(9*x))/2.
a(0)=1, a(1)=18, a(2)=225; for n > 2, a(n) = 18*a(n-1) - 99*a(n-2) + 162*a(n-3). - Vincenzo Librandi, Jul 02 2013
a(n) = 15*a(n-1) - 54*a(n-2) + 3^n. - Vincenzo Librandi, Jul 02 2013

A383935 Expansion of 1 / ( (1-3x) (1-6*x) )^(1/3).

Original entry on oeis.org

1, 3, 12, 54, 261, 1323, 6930, 37152, 202554, 1118286, 6233760, 35014356, 197881866, 1123990182, 6411554028, 36705925656, 210797967321, 1213895891835, 7007131607220, 40534622188830, 234931402041525, 1363961443750155, 7931187074571930, 46183636475060760

Offset: 0

Seiichi Manyama, Aug 18 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A016137, A383937.

Cf. A383627.

R := PowerSeriesRing(Rationals(), 34); f := 1 / ( (1-3*x) * (1-6*x) )^(1/3); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025

CoefficientList[Series[1/((1-3*x)*(1-6*x))^(1/3),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *)

a(n) = (-3)^n*sum(k=0, n, 2^k*binomial(-1/3, k)*binomial(-1/3, n-k));

G.f.: B(x)^(1/3), where B(x) is the g.f. of A016137.
a(n) = (-3)^n * Sum_{k=0..n} 2^k * binomial(-1/3,k) * binomial(-1/3,n-k).
a(n) ~ 2^(n + 1/3) * 3^n / (Gamma(1/3) * n^(2/3)). - Vaclav Kotesovec, Aug 18 2025
D-finite with recurrence: (-9*n-3)*a(n)+(18*n-6)*a(n-1)+(n+1)*a(n+1) = 0. - Georg Fischer, Aug 29 2025

A100852 Triangle read by rows: T(n,k) = 2^k * 3^n, 0 <= k <= n.

Original entry on oeis.org

1, 3, 6, 9, 18, 36, 27, 54, 108, 216, 81, 162, 324, 648, 1296, 243, 486, 972, 1944, 3888, 7776, 729, 1458, 2916, 5832, 11664, 23328, 46656, 2187, 4374, 8748, 17496, 34992, 69984, 139968, 279936, 6561, 13122, 26244, 52488, 104976, 209952, 419904, 839808

Offset: 0

Reinhard Zumkeller, Nov 20 2004

T(n,0) = A000244(n); T(n,n) = A000400(n) = A100851(n,n);
T(n,1) = A008776(n) for n>0;
T(n,2) = A003946(n+1) for n>1;
T(n,3) = A005051(n+1) for n>2;
T(n,n-1) = A081341(n+1) for n>0;
row sums give A016137.

			Triangle begins:
   1;
   3,   6;
   9,  18,  36;
  27,  54, 108, 216;
  81, 162, 324, 648, 1296;
...

Michael De Vlieger, Table of n, a(n) for n = 0..10010 (Rows 0 <= n <= 140).
Eric Weisstein's World of Mathematics, Smooth Number

Cf. A100851, A003586, A065333(T(n, k))=1.

Table[2^k*3^n, {n, 0, 140}, {k, 0, n}] // Flatten (* Michael De Vlieger, Mar 06 2017 *)

for(n=0, 8, for(k=0, n, print1(2^k*3^n", "))) \\ Satish Bysany, Mar 06 2017

G.f.: 1/((1 - 3*x)(1 - 6*x*y)). - Ilya Gutkovskiy, Jun 03 2017

A016765 Expansion of g.f. 1/((1-3x)(1-4x)(1-6*x)).

Original entry on oeis.org

1, 13, 115, 865, 5971, 39193, 249355, 1555105, 9573091, 58428073, 354585595, 2143759345, 12928070611, 77832076153, 468051849835, 2812563019585, 16892428846531, 101422905135433, 608811146458075, 3653962903591825, 21928165007708851, 131586550851237913, 789589579708426315

Offset: 0

N. J. A. Sloane, Dec 11 1999

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-54,72).

Cf. A016137, A016149.

[6^(n+1)-2^(2*n+3)+3^(n+1): n in [0..20]]; // Wesley Ivan Hurt, May 15 2014

A016765:=n->6^(n+1)-2^(2*n+3)+3^(n+1); seq(A016765(n), n=0..20); # Wesley Ivan Hurt, May 15 2014

Table[6^(n + 1) - 2^(2*n + 3) + 3^(n + 1), {n, 0, 20}] (* Wesley Ivan Hurt, May 15 2014 *)
CoefficientList[Series[1/((1-3x)(1-4x)(1-6x)),{x,0,30}],x] (* or *) LinearRecurrence[{13,-54,72},{1,13,115},30] (* Harvey P. Dale, Jul 18 2021 *)

vector(30,n,n--; 6^(n+1)-2^(2*n+3)+3^(n+1)) \\ G. C. Greubel, Sep 15 2018

From Vincenzo Librandi, Mar 20 2011: (Start)
a(n) = 6^(n+1) - 2^(2*n+3) + 3^(n+1).
a(n) = 10*a(n-1) - 24*a(n-2) + 3^n, n >= 2. (End)
G.f.: 1/((1-3*x)*(1-4*x)*(1-6*x)) = -3/(1-3*x) + 8/(1-4*x) - 6/(1-6*x). - Wolfdieter Lang, May 19 2014
From Elmo R. Oliveira, Mar 26 2025: (Start)
E.g.f.: exp(3*x)*(6*exp(3*x) - 8*exp(x) + 3).
a(n) = 13*a(n-1) - 54*a(n-2) + 72*a(n-3).
a(n) = A016149(n+1) - A016137(n+1). (End)

A327317 Triangular array read by rows: row n shows the coefficients of this polynomial of degree n: p(x,n) = 2^(n-1) ((x+r)^n - (x+s)^n)/(r - s), where r = 2 and s = 1/2.

Original entry on oeis.org

1, 5, 4, 21, 30, 12, 85, 168, 120, 32, 341, 850, 840, 400, 80, 1365, 4092, 5100, 3360, 1200, 192, 5461, 19110, 28644, 23800, 11760, 3360, 448, 21845, 87376, 152880, 152768, 95200, 37632, 8960, 1024, 87381, 393210, 786384, 917280, 687456, 342720, 112896

Offset: 1

Clark Kimberling, Nov 03 2019

p(x,n) is a strong divisibility sequence of polynomials. That is, gcd(p(x,h),p(x,k)) = p(x,gcd(h,k)). If x is an integer, then p(x,n) is a strong divisibility sequence of integers.

			First six rows:
     1;
     5,    4;
    21,   30,   12;
    85,  168,  120,   32;
   341,  850,  840,  400,   80;
  1365, 4092, 5100, 3360, 1200, 192;
The first six polynomials, not factored:
1, 5 + 4 x, 21 + 30 x + 12 x^2, 85 + 168 x + 120 x^2 + 32 x^3, 341 + 850 x + 840 x^2 + 400 x^3 + 80 x^4, 1365 + 4092 x + 5100 x^2 + 3360 x^3 + 1200 x^4 + 192 x^5.
The first six polynomials, factored:
1, 5 + 4 x, 3 (7 + 10 x + 4 x^2), (5 + 4 x) (17 + 20 x + 8 x^2), 341 + 850 x + 840 x^2 + 400 x^3 + 80 x^4, 3 (5 + 4 x) (7 + 10 x + 4 x^2) (13 + 10 x + 4 x^2).

Cf. A327316, A002450 (x=0), A016137 (x=1), A001045 (x = -1), A016162 (x = 2), A016181 (x = 3), A016127 (x = -3), A016157 (x = 1/2).

r = 2; s = 1/2; f[x_, n_] := 2^(n - 1) ((x + r)^n - (x + s)^n)/(r - s);
Column[Table[Expand[f[x, n]], {n, 1, 5}]]
c[x_, n_] := CoefficientList[Expand[f[x, n]], x]
TableForm[Table[c[x, n], {n, 1, 10}]] (* A327317 array *)
Flatten[Table[c[x, n], {n, 1, 12}]]   (* A327317 sequence *)

A383937 Expansion of 1 / ( (1-3x) (1-6*x) )^(2/3).

Original entry on oeis.org

1, 6, 33, 180, 990, 5508, 30978, 175824, 1005345, 5782590, 33418737, 193876092, 1128297276, 6583492080, 38498441400, 225550220544, 1323563204394, 7777806812892, 45762197971050, 269545947941160, 1589219394582996, 9378142402189176, 55385341859409948

Offset: 0

Seiichi Manyama, Aug 18 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A016137, A383935.

R := PowerSeriesRing(Rationals(), 34); f := 1 / ( (1-3*x) * (1-6*x) )^(2/3); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025

CoefficientList[Series[1/((1-3*x)*(1-6*x))^(2/3),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *)

a(n) = (-3)^n*sum(k=0, n, 2^k*binomial(-2/3, k)*binomial(-2/3, n-k));

G.f.: B(x)^(2/3), where B(x) is the g.f. of A016137.
a(n) = (-3)^n * Sum_{k=0..n} 2^k * binomial(-2/3,k) * binomial(-2/3,n-k).
a(n) ~ Gamma(1/3) * 2^(n - 1/3) * 3^(n + 1/2) / (Pi * n^(1/3)). - Vaclav Kotesovec, Aug 18 2025
D-finite with recurrence n*a(n) +3*(-3*n+1)*a(n-1) +6*(3*n-2)*a(n-2)=0. - R. J. Mathar, Aug 26 2025

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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A075498 Stirling2 triangle with scaled diagonals (powers of 3).

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A017933 Expansion of 1/((1-3x)(1-6x)(1-9x)).

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A383935 Expansion of 1 / ( (1-3*x) * (1-6*x) )^(1/3).

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A100852 Triangle read by rows: T(n,k) = 2^k * 3^n, 0 <= k <= n.

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A016765 Expansion of g.f. 1/((1-3*x)*(1-4*x)*(1-6*x)).

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A327317 Triangular array read by rows: row n shows the coefficients of this polynomial of degree n: p(x,n) = 2^(n-1) ((x+r)^n - (x+s)^n)/(r - s), where r = 2 and s = 1/2.

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

A383935 Expansion of 1 / ( (1-3x) (1-6*x) )^(1/3).

A016765 Expansion of g.f. 1/((1-3x)(1-4x)(1-6*x)).

A383937 Expansion of 1 / ( (1-3x) (1-6*x) )^(2/3).