cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

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

Original entry on oeis.org

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

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Author

Keywords

Crossrefs

Row sums of A100851.
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).

Programs

Formula

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

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Author

Wolfdieter Lang, Oct 02 2002

Keywords

Comments

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

Examples

			[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)
		

Crossrefs

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

Programs

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

Formula

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

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Author

Keywords

Crossrefs

Third column of triangle A075498.

Programs

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

Formula

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 / ( (1-3*x) * (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

Views

Author

Seiichi Manyama, Aug 18 2025

Keywords

Crossrefs

Programs

  • Magma
    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
  • Mathematica
    CoefficientList[Series[1/((1-3*x)*(1-6*x))^(1/3),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *)
  • PARI
    a(n) = (-3)^n*sum(k=0, n, 2^k*binomial(-1/3, k)*binomial(-1/3, n-k));
    

Formula

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

Views

Author

Reinhard Zumkeller, Nov 20 2004

Keywords

Comments

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.

Examples

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

Crossrefs

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

Programs

  • Mathematica
    Table[2^k*3^n, {n, 0, 140}, {k, 0, n}] // Flatten (* Michael De Vlieger, Mar 06 2017 *)
  • PARI
    for(n=0, 8, for(k=0, n, print1(2^k*3^n", "))) \\ Satish Bysany, Mar 06 2017

Formula

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

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

Views

Author

N. J. A. Sloane, Dec 11 1999

Keywords

Crossrefs

Programs

  • Magma
    [6^(n+1)-2^(2*n+3)+3^(n+1): n in [0..20]]; // Wesley Ivan Hurt, May 15 2014
    
  • Maple
    A016765:=n->6^(n+1)-2^(2*n+3)+3^(n+1); seq(A016765(n), n=0..20); # Wesley Ivan Hurt, May 15 2014
  • Mathematica
    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 *)
  • PARI
    vector(30,n,n--; 6^(n+1)-2^(2*n+3)+3^(n+1)) \\ G. C. Greubel, Sep 15 2018

Formula

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

Views

Author

Clark Kimberling, Nov 03 2019

Keywords

Comments

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.

Examples

			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).
		

Crossrefs

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

Programs

  • Mathematica
    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 / ( (1-3*x) * (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

Views

Author

Seiichi Manyama, Aug 18 2025

Keywords

Crossrefs

Programs

  • Magma
    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
  • Mathematica
    CoefficientList[Series[1/((1-3*x)*(1-6*x))^(2/3),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *)
  • PARI
    a(n) = (-3)^n*sum(k=0, n, 2^k*binomial(-2/3, k)*binomial(-2/3, n-k));
    

Formula

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

Views

Author

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

Keywords

Comments

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.

Examples

			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, ...
  ...
		

Programs

  • PARI
    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())
Showing 1-9 of 9 results.