A132159 -id:A132159 - cp's OEIS frontend

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.

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

Paul D. Hanna, Feb 07 2008

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)

			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.

G. C. Greubel, Rows n=0..100 of triangle, flattened
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3
Wikipedia, Sheffer sequence

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

Cf. A094587.

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 *)

T(n,k)=binomial(n,k)*if(n-k==0,1,prod(j=0,n-k-1,3*j+1))

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)

A324224 Total number T(n,k) of 1's in falling diagonals with index k in all n X n permutation matrices divided by |k|!; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.

Original entry on oeis.org

1, 1, 2, 1, 1, 4, 6, 4, 1, 1, 6, 18, 24, 18, 6, 1, 1, 8, 36, 96, 120, 96, 36, 8, 1, 1, 10, 60, 240, 600, 720, 600, 240, 60, 10, 1, 1, 12, 90, 480, 1800, 4320, 5040, 4320, 1800, 480, 90, 12, 1, 1, 14, 126, 840, 4200, 15120, 35280, 40320, 35280, 15120, 4200, 840, 126, 14, 1

Offset: 1

Alois P. Heinz, Feb 18 2019

			Triangle T(n,k) begins:
  :                                 1                              ;
  :                           1,    2,    1                        ;
  :                     1,    4,    6,    4,    1                  ;
  :               1,    6,   18,   24,   18,    6,   1             ;
  :          1,   8,   36,   96,  120,   96,   36,   8,  1         ;
  :      1, 10,  60,  240,  600,  720,  600,  240,  60, 10,  1     ;
  :  1, 12, 90, 480, 1800, 4320, 5040, 4320, 1800, 480, 90, 12, 1  ;

Alois P. Heinz, Rows n = 1..100, flattened
Wikipedia, Permutation
Wikipedia, Permutation matrix

Columns k=0-6 give (offsets may differ): A000142, A001563, A001286, A005990, A061206, A062199, A062148.
Row sums give A306495(n-1).
Cf. A132159 (right part of triangle), A306234, A324225.

b:= proc(s, c) option remember; (n-> `if`(n=0, c,
      add(b(s minus {i}, c+x^(n-i)), i=s)))(nops(s))
    end:
T:= n-> (p-> seq(coeff(p, x, i)/abs(i)!, i=1-n..n-1))(b({$1..n}, 0)):
seq(T(n), n=1..8);
# second Maple program:
egf:= k-> (t-> x^t/t!*hypergeom([2, t], [t+1], x))(abs(k)+1):
T:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(seq(T(n, k), k=1-n..n-1), n=1..8);
# third Maple program:
T:= (n, k)-> (t-> `if`(t

T[n_, k_] := With[{t = Abs[k]}, If[tJean-François Alcover, Mar 25 2021, after 3rd Maple program *)

T(n,k) = T(n,-k).
T(n,k) = (n-t)*(n-1)!/t! if t < n with t = |k|, T(n,k) = 0 otherwise.
T(n,k) = 1/|k|! * A324225(n,k).
E.g.f. of column k: x^t/t! * hypergeom([2, t], [t+1], x) with t = |k|+1.
Sum_{k=1-n..n-1} T(n,k) = A306495(n-1).

A121757 Triangle read by rows: multiply Pascal's triangle by 1,2,6,24,120,720,... = A000142.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 6, 18, 24, 1, 8, 36, 96, 120, 1, 10, 60, 240, 600, 720, 1, 12, 90, 480, 1800, 4320, 5040, 1, 14, 126, 840, 4200, 15120, 35280, 40320, 1, 16, 168, 1344, 8400, 40320, 141120, 322560, 362880, 1, 18, 216, 2016, 15120, 90720, 423360, 1451520

Offset: 0

Alford Arnold, Aug 19 2006

Row sums are 1,3,11,49,261,1631,... = A001339

a(n,k) = D(n+1,k+1) Array D in A253938 is part of a conjectured formula for F(n,p,r) that relates Dyck path peaks and returns. a(n,k) was discovered prior to array D. - Roger Ford, May 19 2016

			Row 6 is 1*1 5*2 10*6 10*24 5*120 1*720.
From _Vincenzo Librandi_, Dec 16 2012: (Start)
Triangle begins:
1,
1, 2,
1, 4,  6,
1, 6,  18,  24,
1, 8,  36,  96,   120,
1, 10, 60,  240,  600,  720,
1, 12, 90,  480,  1800, 4320,  5040,
1, 14, 126, 840,  4200, 15120, 35280,  40320,
1, 16, 168, 1344, 8400, 40320, 141120, 322560, 362880 etc.
(End)

Vincenzo Librandi, Rows n = 0..100, flattened
J. Goldman, J. Haglund, Generalized rook polynomials, J. Combin. Theory A91 (2000), 509-530, 2-rook coefficients of k rooks on the 2xn board (all heights 2).
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007526 A000522, A005843 (2nd column), A028896 (3rd column).
Cf. A008279.
Cf. A008277, A132159 (mirrored).

a121757 n k = a121757_tabl !! n !! k
a121757_row n = a121757_tabl !! n
a121757_tabl = iterate
   (\xs -> zipWith (+) (xs ++ [0]) (zipWith (*) [1..] ([0] ++ xs))) [1]
-- Reinhard Zumkeller, Mar 06 2014

Flatten[Table[n!(k+1)/(n-k)!,{n,0,10},{k,0,n}]]  (* Harvey P. Dale, Apr 25 2011 *)

A000142(n)={ return(n!) ; } A007318(n,k)={ return(binomial(n,k)) ; } A121757(n,k)={ return(A007318(n,k)*A000142(k+1)) ; } { for(n=0,12, for(k=0,n, print1(A121757(n,k),",") ; ); ) ; } \\ R. J. Mathar, Sep 02 2006

a(n,k) = A007318(n,k)*A000142(k+1), k=0,1,..,n, n=0,1,2,3... - R. J. Mathar, Sep 02 2006

a(n,k) = A008279(n,k) * (k+1). a(n,k) = n!*(k+1)/(n-k)!. - Franklin T. Adams-Watters, Sep 20 2006

A176861 Triangle T(n, k) = (-1)^n(k+1)!(n-k+1)!binomial(n+2, k+2)binomial(n+2, n-k+2) read by rows.

Original entry on oeis.org

1, -6, -6, 36, 64, 36, -240, -600, -600, -240, 1800, 5760, 8100, 5760, 1800, -15120, -58800, -105840, -105840, -58800, -15120, 141120, 645120, 1411200, 1806336, 1411200, 645120, 141120, -1451520, -7620480, -19595520, -30481920, -30481920, -19595520, -7620480, -1451520

Offset: 0

Roger L. Bagula, Apr 27 2010

Row sums are: 1, -12, 136, -1680, 23220, -359520, 6201216, -118298880, ...

			Triangle begins as:
       1;
      -6,     -6;
      36,     64,      36;
    -240,   -600,    -600,    -240;
    1800,   5760,    8100,    5760,    1800;
  -15120, -58800, -105840, -105840,  -58800, -15120;
  141120, 645120, 1411200, 1806336, 1411200, 645120, 141120;

F. S. Roberts, Applied Combinatorics, Prentice-Hall, 1984, p. 576 and 270.

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A132159.

[(-1)^n*Factorial(k+1)*Factorial(n-k+1)*Binomial(n+2, k+2)*Binomial(n+2, n-k+2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 07 2021

T[n_, k_]:= (-1)^n*(k+1)!*(n-k+1)!*Binomial[n+2, k+2]*Binomial[n+2, n-k+2];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

flatten([[(-1)^n*factorial(k+1)*factorial(n-k+1)*binomial(n+2, k+2)*binomial(n+2, n-k+2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 07 2021

T(n, k) = (-1)^n*(k+1)!*(n-k+1)!*binomial(n+2, k+2)*binomial(n+2, n-k+2).

T(n, k) = (-1)^n * A132159(n+2, k+2) * A132159(n+2, n-k+2). - G. C. Greubel, Feb 07 2021

Edited by G. C. Greubel, Feb 07 2021

A176862 A symmetrical triangle sequence:t(n,m)=(-1)^m(n - m)!Binomial[n - 1, m] + (-1)^(n - m + 1)(n - (n - m + 1))!Binomial[n - 1, n - m + 1].

Original entry on oeis.org

2, 5, 5, 37, -16, 37, 239, -50, -50, 239, 1801, -492, 180, -492, 1801, 15119, -4186, 714, 714, -4186, 15119, 141121, -40336, 8568, -2688, 8568, -40336, 141121, 1451519, -423342, 90504, -13104, -13104, 90504, -423342, 1451519

Offset: 3

Roger L. Bagula, Apr 27 2010

Row sums are:

{2, 10, 58, 378, 2798, 23294, 216018, 2211154,...}.

			{2},
{5, 5},
{37, -16, 37},
{239, -50, -50, 239},
{1801, -492, 180, -492, 1801},
{15119, -4186, 714, 714, -4186, 15119},
{141121, -40336, 8568, -2688, 8568, -40336, 141121},
{1451519, -423342, 90504, -13104, -13104, 90504, -423342, 1451519}

F. S. Roberts, Applied Combinatorics, Prentice-Hall, 1984, p. 576 and 270.

Cf. A132159

t[n_, m_] := (-1)^m*(n - m)!*Binomial[n - 1, m] + (-1)^(n - m + 1)*(n - (n - m + 1))!*Binomial[n - 1, n - m + 1];
Table[Table[t[n, m], {m, 2, n - 1}], {n, 3, 10}];
Flatten[%]

t(n,m)=(-1)^m*(n - m)!*Binomial[n - 1, m] + (-1)^(n - m + 1)*(n - (n - m + 1))!*Binomial[n - 1, n - m + 1]

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A324224 Total number T(n,k) of 1's in falling diagonals with index k in all n X n permutation matrices divided by |k|!; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A121757 Triangle read by rows: multiply Pascal's triangle by 1,2,6,24,120,720,... = A000142.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A176861 Triangle T(n, k) = (-1)^n*(k+1)!*(n-k+1)!*binomial(n+2, k+2)*binomial(n+2, n-k+2) read by rows.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

A176862 A symmetrical triangle sequence:t(n,m)=(-1)^m*(n - m)!*Binomial[n - 1, m] + (-1)^(n - m + 1)*(n - (n - m + 1))!*Binomial[n - 1, n - m + 1].

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

Formula

A176861 Triangle T(n, k) = (-1)^n(k+1)!(n-k+1)!binomial(n+2, k+2)binomial(n+2, n-k+2) read by rows.

A176862 A symmetrical triangle sequence:t(n,m)=(-1)^m(n - m)!Binomial[n - 1, m] + (-1)^(n - m + 1)(n - (n - m + 1))!Binomial[n - 1, n - m + 1].