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 1-5 of 5 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

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

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).

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

A075501 Stirling2 triangle with scaled diagonals (powers of 6).

Original entry on oeis.org

1, 6, 1, 36, 18, 1, 216, 252, 36, 1, 1296, 3240, 900, 60, 1, 7776, 40176, 19440, 2340, 90, 1, 46656, 489888, 390096, 75600, 5040, 126, 1, 279936, 5925312, 7511616, 2204496, 226800, 9576, 168, 1, 1679616, 71383680
Offset: 1

Views

Author

Wolfdieter Lang, Oct 02 2002

Keywords

Comments

This is a lower triangular infinite matrix of the Jabotinsky type. See the 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(6*z) - 1)*x/6) - 1.

Examples

			[1]; [6,1]; [36,18,1]; ...; p(3,x) = x(36 + 18*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*      1
*      6       1
*     36      18       1
*    216     252      36       1
*   1296    3240     900      60      1
*   7776   40176   19440    2340     90    1
*  46656  489888  390096   75600   5040  126   1
* 279936 5925312 7511616 2204496 226800 9576 168 1
(End)

Links

Crossrefs

Columns 1-7 are A000400, A016175, A075916-A075920. Row sums are A005012.
Cf. A075500, A075502.

Programs

  • Maple
    # The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> 6^n, 9); # Peter Luschny, Jan 28 2016
  • Mathematica
    Flatten[Table[6^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 10;
M = BellMatrix[6^#&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
  • PARI
    for(n=1, 11, for(m=1, n, print1(6^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

Formula

a(n, m) = (6^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*6)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 6m*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}(1-6k*x), m >= 1.
E.g.f. for m-th column: (((exp(6x)-1)/6)^m)/m!, m >= 1.

A075916 Third column of triangle A075501.

Original entry on oeis.org

1, 36, 900, 19440, 390096, 7511616, 141134400, 2611802880, 47870735616, 871982724096, 15819463296000, 286235993272320, 5170077903015936, 93275375604350976, 1681524519443251200, 30298254922942709760
Offset: 0

Views

Author

Wolfdieter Lang, Oct 02 2002

Keywords

Comments

The e.g.f. given below is Sum_{m=0..2} A075513(3,m)*exp(6*(m+1)*x)/2!.

Crossrefs

Cf. A016175, A075917.

Formula

a(n) = A075501(n+3, 3) = (6^n)*S2(n+3, 3) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = (6^n - 8*12^n + 9*18^n)/2.
G.f.: 1/Product_{k=1..3} (1 - 6*k*x).
E.g.f.: (d^3/dx^3)(((exp(6*x)-1)/6)^3)/3! = (exp(6*x) - 8*exp(12*x) + 9*exp(18*x))/2!.

A361290 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..floor((n-1)/2)} k^(n-1-j) * binomial(n,2*j+1).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 2, 0, 0, 1, 4, 4, 0, 0, 1, 6, 14, 8, 0, 0, 1, 8, 30, 48, 16, 0, 0, 1, 10, 52, 144, 164, 32, 0, 0, 1, 12, 80, 320, 684, 560, 64, 0, 0, 1, 14, 114, 600, 1936, 3240, 1912, 128, 0, 0, 1, 16, 154, 1008, 4400, 11648, 15336, 6528, 256, 0
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2023

Keywords

Examples

			Square array begins:
  0,  0,   0,   0,    0,    0, ...
  1,  1,   1,   1,    1 ,   1, ...
  0,  2,   4,   6,    8,   10, ...
  0,  4,  14,  30,   52,   80, ...
  0,  8,  48, 144,  320,  600, ...
  0, 16, 164, 684, 1936, 4400, ...

Crossrefs

Column k=1..10 give A131577, A007070(n-1), A030192(n-1), A016129(n-1), A093145, A154237, A154248, A154348(n-1), A016175(n-1), A361293.
Main diagonal gives A360766.
Cf. A361432.

Programs

  • PARI
    T(n, k) = polcoef(lift(Mod('x, ('x-k)^2-k)^n), 1);

Formula

T(0,k) = 0, T(1,k) = 1; T(n,k) = 2 * k * T(n-1,k) - (k-1) * k * T(n-2,k).
T(n,k) = ((k + sqrt(k))^n - (k - sqrt(k))^n)/(2 * sqrt(k)) for k > 0.
G.f. of column k: x/(1 - 2 * k * x + (k-1) * k * x^2).
E.g.f. of column k: exp(k * x) * sinh(sqrt(k) * x) / sqrt(k) for k > 0.

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

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016174, A016175.

Programs

  • Mathematica
    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 *)

Formula

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)

Extensions

More terms from Elmo R. Oliveira, Mar 26 2025
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.