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 1-6 of 6 results.

A112334 Inverse of number triangle A112333.

Original entry on oeis.org

1, -2, 1, 0, -5, 1, 0, 0, -8, 1, 0, 0, 0, -11, 1, 0, 0, 0, 0, -14, 1, 0, 0, 0, 0, 0, -17, 1, 0, 0, 0, 0, 0, 0, -20, 1, 0, 0, 0, 0, 0, 0, 0, -23, 1, 0, 0, 0, 0, 0, 0, 0, 0, -26, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -29, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -32, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -35, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -38, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Paul Barry, Sep 04 2005

Keywords

Comments

Row sums are A112335.

Examples

			Triangle begins
1;
-2,1;
0,-5,1;
0,0,-8,1;
0,0,0,-11,1;
0,0,0,0,-14,1;
0,0,0,0,0,-17,1;

A136214 Triangle U, read by rows, where U(n,k) = Product_{j=k..n-1} (3*j+1) for n > k with U(n,n) = 1.

Original entry on oeis.org

1, 1, 1, 4, 4, 1, 28, 28, 7, 1, 280, 280, 70, 10, 1, 3640, 3640, 910, 130, 13, 1, 58240, 58240, 14560, 2080, 208, 16, 1, 1106560, 1106560, 276640, 39520, 3952, 304, 19, 1, 24344320, 24344320, 6086080, 869440, 86944, 6688, 418, 22, 1
Offset: 0

Views

Author

Paul D. Hanna, Feb 07 2008

Keywords

Comments

Let G(m, k, p) = (-p)^k*Product_{j=0..k-1}(j - m - 1/p) and T(n, k, p) = G(n-1, n-k, p) then T(n, k, 1) = A094587(n, k), T(n, k, 2) = A112292(n, k) and T(n, k, 3) is this sequence. - Peter Luschny, Jun 01 2009, revised Jun 18 2019

Examples

			Triangle begins:
        1;
        1,       1;
        4,       4,      1;
       28,      28,      7,     1;
      280,     280,     70,    10,    1;
     3640,    3640,    910,   130,   13,   1;
    58240,   58240,  14560,  2080,  208,  16,  1;
  1106560, 1106560, 276640, 39520, 3952, 304, 19, 1; ...
Matrix inverse begins:
   1;
  -1,   1;
   0,  -4,   1;
   0,   0,  -7,   1;
   0,   0,   0, -10,   1;
   0,   0,   0,   0, -13,   1; ...

Links

Crossrefs

Cf. A094587, A112333, A136216, A136239; A007559, A136212, A136213.

Programs

  • Magma
    [[n eq 0 select 1 else k eq n select 1 else (&*[3*j+1: j in [k..n-1]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Jun 14 2019
  • Maple
    nmax:=8; for n from 0 to nmax do U(n, n):=1 od: for n from 0 to nmax do for k from 0 to n do if n > k then U(n, k) := mul((3*j+1), j = k..n-1) fi: od: od: for n from 0 to nmax do seq(U(n, k), k=0..n) od: seq(seq(U(n, k), k=0..n), n=0..nmax); # Johannes W. Meijer, Jul 04 2011, revised Nov 23 2012
  • Mathematica
    Table[Product[3*j+1, {j,k,n-1}], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 14 2019 *)
  • PARI
    T(n,k)=if(n==k,1,prod(j=k,n-1,3*j+1))
  • Sage
    def T(n, k):
    if (k==n): return 1
    else: return product(3*j+1 for j in (k..n-1))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jun 14 2019

Formula

Matrix powers: column 0 of U^(k+1) = column k of A136216 for k >= 0; simultaneously, column k = column 0 of A136216^(3k+1) for k >= 0. Element in column 0, row n, of matrix power U^(k+1) = A007559(n)*C(n+k,k), where A007559 are triple factorials found in column 0 of this triangle.

A136215 Triangle T, read by rows, where T(n,k) = A007559(n-k)*C(n,k) where A007559 equals the triple factorials in column 0.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 28, 12, 3, 1, 280, 112, 24, 4, 1, 3640, 1400, 280, 40, 5, 1, 58240, 21840, 4200, 560, 60, 6, 1, 1106560, 407680, 76440, 9800, 980, 84, 7, 1, 24344320, 8852480, 1630720, 203840, 19600, 1568, 112, 8, 1, 608608000, 219098880, 39836160
Offset: 0

Views

Author

Paul D. Hanna, Feb 07 2008

Keywords

Comments

Comments from Peter Bala, Jul 10 2008: (Start) This array is the particular case P(1,3) of the generalized Pascal triangle P(a,b), a lower unit triangular matrix, shown below
n\k|0....................1...............2.........3.....4
----------------------------------------------------------
0..|1.....................................................
1..|a....................1................................
2..|a(a+b)...............2a..............1................
3..|a(a+b)(a+2b).........3a(a+b).........3a........1......
4..|a(a+b)(a+2b)(a+3b)...4a(a+b)(a+2b)...6a(a+b)...4a....1
...
See A094587 for some general properties of these arrays.
Other cases recorded in the database include: P(1,0) = Pascal's triangle A007318, P(1,1) = A094587, P(2,0) = A038207, P(3,0) = A027465, P(2,1) = A132159 and P(2,3) = A136216. (End)
The generalized Pascal matrix that Bala refers to is itself a special case of application of the formalism of A133314 to fundamental matrices derived from infinitesimal generators described in A133314, of which the fundamental Pascal (A007318), unsigned Lah (A105278) and associated Laguerre (A135278) matrices are special examples. The formalism gives, among other relations, the inverse of T as TI(n,k) = b(n-k)*C(n,k) where the sequence b is given by the list partition transform (A133314) of A007559; i.e., b = LPT(A007559) = (1,-A008544)= (1,-1,-2,-10,-80,...). The formalism of A132382 may also be applied with the double factorial A001147 replaced by the triple factorial A007559 (see also A133480). - Tom Copeland, Aug 18 2008
From Peter Bala, Aug 29 2013: (Start)
Exponential Riordan array [1/(1 - 3*y)^(1/3), y]. The row polynomials R(n,x) thus form a Sheffer sequence of polynomials with associated delta operator equal to d/dx. Thus d/dx(R(n,x)) = n*R(n-1,x). The Sheffer identity is R(n,x + y) = sum {k = 0..n} binomial(n,k)*y^(n-k)*R(k,x).
Define a polynomial sequence P(n,x) of binomial type by setting P(n,x) = product {k = 0..n-1} (x + 3*k) with the convention that P(0,x) = 1. Then this is triangle of connection constants when expressing the basis polynomials P(n,x + 1) in terms of the basis P(n,x).
For example, row 3 is (28, 12, 3, 1) so P(3,x + 1) = (x + 1)*(x + 4)*(x + 7) = 28 + 12*x + 3*x*(x + 3) + x*(x + 3)*(x + 6). (End)

Examples

			Column k of T = column 0 of U^(k+1), while
column k of U = column 0 of T^(3k+1) where U = A136214 and
column k of V = column 0 of T^(3k+2) where V = A112333.
This triangle T begins:
        1;
        1,      1;
        4,      2,     1;
       28,     12,     3,    1;
      280,    112,    24,    4,   1;
     3640,   1400,   280,   40,   5,  1;
    58240,  21840,  4200,  560,  60,  6, 1;
  1106560, 407680, 76440, 9800, 980, 84, 7, 1; ...
Triangle U = A136214 begins:
     1;
     1,    1;
     4,    4,   1;
    28,   28,   7,   1;
   280,  280,  70,  10,  1;
  3640, 3640, 910, 130, 13, 1; ...
with triple factorials A007559 in column 0.
Triangle V = A112333 begins:
      1;
      2,    1;
     10,    5,    1;
     80,   40,    8,   1;
    880,  440,   88,  11,  1;
  12320, 6160, 1232, 154, 14, 1; ...
with triple factorials A008544 in column 0.

Links

Crossrefs

Cf. A136216 (matrix square); A007559, A008544; A136212, A136213.
Cf. A094587.

Programs

  • Mathematica
    T[n_, k_]:= Binomial[n, k]*If[n - k == 0, 1, Product[3*j + 1, {j, 0, n - k - 1}]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 10 2018 *)
  • PARI
    T(n,k)=binomial(n,k)*if(n-k==0,1,prod(j=0,n-k-1,3*j+1))

Formula

Column k of T = column 0 of U^(k+1) (matrix power) for k>=0 where U = A136214. Matrix square equals A136216, where A136216(n,k) = A008544(n-k)*C(n,k) where A008544 are also triple factorials.
From Peter Bala, Jul 10 2008: (Start)
T(n,k) = (3*n-3*k-2)*T(n-1,k) + T(n-1,k-1).
E.g.f. exp(x*y)/(1-3*y)^(1/3) = 1 + (1+x)*y + (4+2*x+x^2)*y^2/2! + ... . (End)

A136216 Triangle T, read by rows, where T(n,k) = A008544(n-k)*C(n,k) where A008544 equals the triple factorials in column 0.

Original entry on oeis.org

1, 2, 1, 10, 4, 1, 80, 30, 6, 1, 880, 320, 60, 8, 1, 12320, 4400, 800, 100, 10, 1, 209440, 73920, 13200, 1600, 150, 12, 1, 4188800, 1466080, 258720, 30800, 2800, 210, 14, 1, 96342400, 33510400, 5864320, 689920, 61600, 4480, 280, 16, 1
Offset: 0

Views

Author

Paul D. Hanna, Feb 07 2008

Keywords

Comments

This array is the particular case P(2,3) of the generalized Pascal triangle P(a,b), a lower unit triangular matrix, shown in the comments to A094587. - Peter Bala, Jul 10 2008
The row polynomials form an Appell sequence. - Tom Copeland, Dec 03 2013

Examples

			Triangle begins:
1;
2, 1;
10, 4, 1;
80, 30, 6, 1;
880, 320, 60, 8, 1;
12320, 4400, 800, 100, 10, 1;
209440, 73920, 13200, 1600, 150, 12, 1;
4188800, 1466080, 258720, 30800, 2800, 210, 14, 1; ...

Links

Crossrefs

Cf. A136215 (square-root), A112333, A008544, A136212, A136213.
Cf. A094587.

Programs

  • Mathematica
    (* The function RiordanArray is defined in A256893. *)
RiordanArray[1/(1 - 3 #)^(2/3)&, #&, 9, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)
  • PARI
    {T(n,k) = binomial(n,k)*if(n-k==0,1, prod(j=0,n-k-1,3*j+2))}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

Formula

Column k of T = column 0 of V^(k+1) for k>=0 where V = A112333.
Equals the matrix square of triangle A136215.
T(n,k) = (3*n-3*k-1)*T(n-1,k) + T(n-1,k-1). - Peter Bala, Jul 10 2008
Using the formalism of A132382 modified for the triple rather than the double factorial (replace 2 by 3 in basic formulas), the e.g.f. for the row polynomials is exp(x*t)*(1-3x)^(-2/3). - Tom Copeland, Aug 18 2008
From Peter Bala, Aug 28 2013: (Start)
Exponential Riordan array [1/(1 - 3*y)^(2/3), y].
The row polynomials R(n,x) thus form a Sheffer sequence of polynomials with associated delta operator equal to d/dx. Thus d/dx(R(n,x)) = n*R(n-1,x). The Sheffer identity is R(n,x + y) = sum {k = 0..n} binomial(n,k)*y^(n-k)*R(k,x).
Define a polynomial sequence P(n,x) of binomial type by setting P(n,x) = product {k = 0..n-1} (2*x + 3*k) with the convention that P(0,x) = 1. Then this is triangle of connection constants when expressing the basis polynomials P(n,x + 1) in terms of the basis P(n,x). For example, row 3 is (80, 30, 6, 1) so P(3,x + 1) = (2*x + 2)*(2*x + 5)*(2*x + 8) = 80 + 20*(2*x) + 6*(2*x*(2*x + 3)) + (2*x)*(2*x + 3)*(2*x + 6). (End)

A190903 a(n) = Product_{k in M_n} k, M_n = {k | 1 <= k <= 3n and k mod 3 = n mod 3}.

Original entry on oeis.org

1, 1, 10, 162, 280, 12320, 524880, 1106560, 96342400, 7142567040, 17041024000, 2324549427200, 254561089305600, 664565853952000, 126757680265216000, 18763697892715776000, 52580450364682240000, 13106744139423334400000, 2480410751833883860992000
Offset: 0

Views

Author

Peter Luschny, Jul 03 2011

Keywords

Comments

For n > 0:
a(3*n) = A032031(3*n) = 3^(3*n) * Gamma(3*n + 1).
a(3*n-1) = A008544(3*n-1) = 3^(3*n-1) * Gamma(3*n - 1/3) / Gamma(2/3).
a(3*n+1) = A007559(3*n+1) = 3^(3*n+3/2) * Gamma(3*n + 4/3) * Gamma(2/3) / (2*Pi).

Links

Crossrefs

Cf. A190901.

Programs

  • Maple
    A190903 := proc(n) local k; mul(k, k = select(k-> k mod 3 = n mod 3, [$1 .. 3*n])) end: seq(A190903(n), n=0..17);
  • Mathematica
    a[n_] := Switch[Mod[n, 3], 0, 3^n Gamma[n+1], 2, 3^n Gamma[n+2/3]/ Gamma[2/3], 1, 3^(n-1) Gamma[n+1/3]/Gamma[4/3]] // Round;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 25 2019 *)
  • PARI
    a(n) = prod(k=1, 3*n, if (k % 3 == n % 3, k, 1)); \\ Michel Marcus, Jun 25 2019 and May 14 2020

Formula

From Johannes W. Meijer, Jul 04 2011: (Start)
a(3*n+3)/(a(3*n)*a(3)) = A006566(n+1); Dodecahedral numbers
a(3*n+4)/a(3*n+1) = A136214(3*n+4, 3*n+1)
a(3*n+5)/a(3*n+2) = A112333(3*n+5, 3*n+2) (End)

A368791 a(n) = A008544(n) * Sum_{k=0..n} 1/A008544(k).

Original entry on oeis.org

1, 3, 16, 129, 1420, 19881, 337978, 6759561, 155469904, 4042217505, 117224307646, 3751177844673, 131291224563556, 4989066533415129, 204551727870020290, 9000276026280892761, 423012973235201959768, 21150648661760097988401, 1120984379073285193385254
Offset: 0

Views

Author

Seiichi Manyama, Jan 05 2024

Keywords

Crossrefs

Row sums of A112333.
Cf. A008544, A368790.

Programs

  • PARI
    a008544(n) = prod(k=1, n, 3*k-1);
a(n) = a008544(n)*sum(k=0, n, 1/a008544(k));

Formula

a(n) = (3*n-1) * a(n-1) + 1.
Showing 1-6 of 6 results.

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