A094587 -id:A094587 - cp's OEIS frontend

A371077 Square array read by ascending antidiagonals: A(n, k) = 3^n*Pochhammer(k/3, n).

1, 0, 1, 0, 1, 1, 0, 4, 2, 1, 0, 28, 10, 3, 1, 0, 280, 80, 18, 4, 1, 0, 3640, 880, 162, 28, 5, 1, 0, 58240, 12320, 1944, 280, 40, 6, 1, 0, 1106560, 209440, 29160, 3640, 440, 54, 7, 1, 0, 24344320, 4188800, 524880, 58240, 6160, 648, 70, 8, 1

Offset: 0

Werner Schulte and Peter Luschny, Mar 10 2024

			The array starts:
  [0] 1,    1,     1,     1,     1,      1,      1,      1,      1, ...
  [1] 0,    1,     2,     3,     4,      5,      6,      7,      8, ...
  [2] 0,    4,    10,    18,    28,     40,     54,     70,     88, ...
  [3] 0,   28,    80,   162,   280,    440,    648,    910,   1232, ...
  [4] 0,  280,   880,  1944,  3640,   6160,   9720,  14560,  20944, ...
  [5] 0, 3640, 12320, 29160, 58240, 104720, 174960, 276640, 418880, ...
.
Seen as the triangle T(n, k) = A(n - k, k):
  [0] 1;
  [1] 0,       1;
  [2] 0,       1,      1;
  [3] 0,       4,      2,     1;
  [4] 0,      28,     10,     3,    1;
  [5] 0,     280,     80,    18,    4,   1;
  [6] 0,    3640,    880,   162,   28,   5,  1;
  [7] 0,   58240,  12320,  1944,  280,  40,  6, 1;
  [8] 0, 1106560, 209440, 29160, 3640, 440, 54, 7, 1;
.
Illustrating the LU decomposition of A:
    / 1                \   / 1 1 1 1 1 ... \   / 1   1   1    1    1 ... \
    | 0   1            |   |   1 2 3 4 ... |   | 0   1   2    3    4 ... |
    | 0   4   2        | * |     1 3 6 ... | = | 0   4  10   18   28 ... |
    | 0  28  24   6    |   |       1 4 ... |   | 0  28  80  162  280 ... |
    | 0 280 320 144 24 |   |         1 ... |   | 0 280 880 1944 3640 ... |
    | . . .            |   | . . .         |   | . . .                   |

Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).

Family m^n*Pochhammer(k/m, n): A094587 (m=1), A370419 (m=2), this sequence (m=3), A370915 (m=4).
Rows: A000012, A001477, A028552.
Columns: A000007, A007559, A008544, A032031, A007559 (shifted), A034000, A034001, A051604, A051605, A051606, A051607, A051608, A051609.
Cf. A303486 (main diagonal), A371079 (row sums of triangle), A371076, A371080.

A := (n, k) -> 3^n*pochhammer(k/3, n):
A := (n, k) -> local j; mul(3*j + k, j = 0..n-1):
# Read by antidiagonals:
T := (n, k) -> A(n - k, k): seq(seq(T(n, k), k = 0..n), n = 0..9);
seq(lprint([n], seq(T(n, k), k = 0..n)), n = 0..9);
# Using the generating polynomials of the rows:
P := (n, x) -> local k; add(Stirling1(n, k)*(-3)^(n - k)*x^k, k=0..n):
seq(lprint([n], seq(P(n, k), k = 0..9)), n = 0..5);
# Using the exponential generating functions of the columns:
EGFcol := proc(k, len) local egf, ser, n; egf := (1 - 3*x)^(-k/3);
ser := series(egf, x, len+2): seq(n!*coeff(ser, x, n), n = 0..len) end:
seq(lprint([k], EGFcol(k, 8)), k = 0..6);
# As a matrix product:
with(LinearAlgebra):
L := Matrix(7, 7, (n, k) -> A371076(n - 1,  k - 1)):
U := Matrix(7, 7, (n, k) -> binomial(n - 1, k - 1)):
MatrixMatrixMultiply(L, Transpose(U));

Table[3^(n-k)*Pochhammer[k/3, n-k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Mar 14 2024 *)

def A(n, k): return 3**n * rising_factorial(k/3, n)
def A(n, k): return (-3)**n * falling_factorial(-k/3, n)

A(n, k) = Product_{j=0..n-1} (3*j + k).
A(n, k) = A(n+1, k-3) / (k - 3) for k > 3.
A(n, k) = Sum_{j=0..n} Stirling1(n, j)*(-3)^(n - j)* k^j.
A(n, k) = k! * [x^k] (exp(x) * p(n, x)), where p(n, x) are the row polynomials of A371080.
E.g.f. of column k: (1 - 3*t)^(-k/3).
E.g.f. of row n: exp(x) * (Sum_{k=0..n} A371076(n, k) * x^k / (k!)).
Sum_{n>=0, k>=0} A(n, k) * x^k * t^n / (n!) = 1/(1 - x/(1 - 3*t)^(1/3)).
Sum_{n>=0, k>=0} A(n, k) * x^k * t^n /(n! * k!) = exp(x/(1 - 3*t)^(1/3)).
The LU decomposition of this array is given by the upper triangular matrix U which is the transpose of A007318 and the lower triangular matrix L = A371076, i.e., A(n, k) = Sum_{i=0..k} A371076(n, i) * binomial(k, i).

A112295 Inverse of a double factorial related triangle.

Original entry on oeis.org

1, -1, 1, 0, -3, 1, 0, 0, -5, 1, 0, 0, 0, -7, 1, 0, 0, 0, 0, -9, 1, 0, 0, 0, 0, 0, -11, 1, 0, 0, 0, 0, 0, 0, -13, 1, 0, 0, 0, 0, 0, 0, 0, -15, 1, 0, 0, 0, 0, 0, 0, 0, 0, -17, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -19, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -21, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -23, 1

Offset: 0

Paul Barry, Sep 01 2005

Inverse of A112292. Similar results can be obtained for higher factorials.

			Triangle begins
   1;
  -1,  1;
   0, -3,  1;
   0,  0, -5,  1;
   0,  0,  0, -7,  1;
   0,  0,  0,  0, -9,   1;
   0,  0,  0,  0,  0, -11,  1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A094587, A134081.

A112295:= func< n,k | k eq n select 1 else k eq n-1 select 1-2*n else 0 >;
[A112295(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Feb 17 2021

T[n_, k_]:= If[k==n, 1, If[k==n-1, 1-2*n, 0]];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 17 2021 *)

def A112295(n,k): return 1 if k==n else 1-2*n if k==n-1 else 0
flatten([[A112295(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Feb 17 2021

From G. C. Greubel, Feb 17 2021: (Start)
T(n, k) = 1 - 2*n if k = n-1 otherwise 0, with T(n, n) = 1.
Sum_{k=0..n} T(n, k) = 1 - 2*n - [n=0]. (End)

A162995 A scaled version of triangle A162990.

Original entry on oeis.org

1, 3, 1, 12, 4, 1, 60, 20, 5, 1, 360, 120, 30, 6, 1, 2520, 840, 210, 42, 7, 1, 20160, 6720, 1680, 336, 56, 8, 1, 181440, 60480, 15120, 3024, 504, 72, 9, 1, 1814400, 604800, 151200, 30240, 5040, 720, 90, 10, 1

Offset: 1

Johannes W. Meijer, Jul 27 2009

We get this scaled version of triangle A162990 by dividing the coefficients in the left hand columns by their 'top-values' and then taking the square root.
T(n,k) = A173333(n+1,k+1), 1 <= k <= n. - Reinhard Zumkeller, Feb 19 2010
T(n,k) = A094587(n+1,k+1), 1 <= k <= n. - Reinhard Zumkeller, Jul 05 2012

			The first few rows of the triangle are:
[1]
[3, 1]
[12, 4, 1]
[60, 20, 5, 1]

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened

Cf. A094587.
A056542(n) equals the row sums for n>=1.
A001710, A001715, A001720, A001725, A001730, A049388, A049389, A049398, A051431 are related to the left hand columns.
A000012, A009056, A002378, A007531, A052762, A052787, A053625 and A159083 are related to the right hand columns.

a162995 n k = a162995_tabl !! (n-1) !! (k-1)
a162995_row n = a162995_tabl !! (n-1)
a162995_tabl = map fst $ iterate f ([1], 3)
   where f (row, i) = (map (* i) row ++ [1], i + 1)
-- Reinhard Zumkeller, Jul 04 2012

a := proc(n, m): (n+1)!/(m+1)! end: seq(seq(a(n, m), m=1..n), n=1..9); # Johannes W. Meijer, revised Nov 23 2012

Table[(n+1)!/(m+1)!, {n, 10}, {m, n}] (* Paolo Xausa, Mar 31 2024 *)

a(n,m) = (n+1)!/(m+1)! for n = 1, 2, 3, ..., and m = 1, 2, ..., n.

A122851 Number triangle T(n,k) = C(k,n-k)*(n-k)!.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 2, 3, 1, 0, 0, 0, 6, 4, 1, 0, 0, 0, 6, 12, 5, 1, 0, 0, 0, 0, 24, 20, 6, 1, 0, 0, 0, 0, 24, 60, 30, 7, 1, 0, 0, 0, 0, 0, 120, 120, 42, 8, 1, 0, 0, 0, 0, 0, 120, 360, 210, 56, 9, 1, 0, 0, 0, 0, 0, 0, 720, 840, 336, 72, 10, 1

Offset: 0

Paul Barry, Sep 14 2006

Row sums are A122852.

Triangle T(n,k), read by rows, given by (0,1,-1,0,0,1,-1,0,0,1,-1,0,0,1,...) DELTA (1,0,0,-1,2,0,0,-2,3,0,0,-3,4,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 12 2011

			Triangle begins
  1;
  0, 1;
  0, 1, 1;
  0, 0, 2, 1;
  0, 0, 2, 3,  1;
  0, 0, 0, 6,  4,  1;
  0, 0, 0, 6, 12,  5, 1;
  0, 0, 0, 0, 24, 20, 6, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A094587, A242653.

T(2n,n) gives A000142.

/* As triangle: */ [[Binomial(k,n-k)*Factorial(n-k): k in [0..n]]: n in [0.. 7]]; // Vincenzo Librandi, Apr 24 2015

Flatten[Table[Binomial[k,n-k](n-k)!,{n,0,10},{k,0,n}]] (* Harvey P. Dale, May 16 2012 *)

Number triangle T(n,k) = [k<=n]*k!/(2k-n)!.

T(n,k) = A008279(k,n-k). - Danny Rorabaugh, Apr 23 2015

A135902 Triangle T, read by rows, where column k of T = column 0 of T^(k+1) for k>0, with column 0 of T = column 0 of T^3 shift right.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 15, 8, 3, 1, 102, 47, 15, 4, 1, 860, 356, 102, 24, 5, 1, 8548, 3252, 860, 186, 35, 6, 1, 97094, 34448, 8548, 1736, 305, 48, 7, 1, 1234324, 412546, 97094, 18754, 3130, 465, 63, 8, 1, 17302880, 5488222, 1234324, 228658, 36630, 5212, 672, 80, 9, 1

Offset: 0

Paul D. Hanna, Dec 18 2007

This is a variant of permutation triangle P = A094587, A094587(n,k) = n!/k!, which may be defined by: triangular matrix P where column k of P = column 0 of P^(k+1), with column 0 of P = column 0 of P^2 shift right.

			Triangle T begins:
1;
1, 1;
3, 2, 1;
15, 8, 3, 1;
102, 47, 15, 4, 1;
860, 356, 102, 24, 5, 1;
8548, 3252, 860, 186, 35, 6, 1;
97094, 34448, 8548, 1736, 305, 48, 7, 1;
1234324, 412546, 97094, 18754, 3130, 465, 63, 8, 1;
17302880, 5488222, 1234324, 228658, 36630, 5212, 672, 80, 9, 1; ...
where column k of T = column 0 of T^(k+1)
with column 0 of T = column 0 of T^3 shift right.
Matrix square of T, T^2, begins:
1;
2, 1;
8, 4, 1;
47, 22, 6, 1;
356, 156, 42, 8, 1;
3252, 1343, 351, 68, 10, 1;
34448, 13493, 3415, 656, 100, 12, 1; ...
where column 0 of T^2 = column 1 of T.
Matrix cube of T, T^3, begins:
1;
3, 1;
15, 6, 1;
102, 42, 9, 1;
860, 351, 81, 12, 1;
8548, 3415, 807, 132, 15, 1;
97094, 37795, 8967, 1530, 195, 18, 1; ...
where column 0 of T^3 = column 2 of T = column 0 of T shift left;
also, column 1 of T^3 = column 2 of T^2.

Cf. columns: A135903, A135904, A135905; variants: A091351, A094587.

T(n, k)=if(k>n || n<0 || k<0, 0, if(k==n,1,if(k==0, T(n+1,2), sum(j=0, n-k, T(n-k, j)*T(j+k-1, k-1))); ); )

Column k of T^(j+1) = column j of T^(k+1) for j>=0, k>=0.

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

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)

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

Paul D. Hanna, Feb 07 2008

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

			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; ...

Wikipedia, Appell sequence
Wikipedia, Sheffer sequence

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

Cf. A094587.

(* The function RiordanArray is defined in A256893. *)
RiordanArray[1/(1 - 3 #)^(2/3)&, #&, 9, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

{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(""))

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)

A138271 Triangle T, read by rows, where column k of T = column 0 of T^(k+1) for k>0, with column 0 of T = column 0 of T^4 shift right.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 28, 10, 3, 1, 268, 78, 18, 4, 1, 3164, 798, 156, 28, 5, 1, 43672, 9874, 1714, 268, 40, 6, 1, 682632, 141282, 22368, 3164, 420, 54, 7, 1, 11834536, 2273730, 333910, 43672, 5320, 618, 70, 8, 1, 224283416, 40400466, 5566728, 682632, 77720

Offset: 0

Paul D. Hanna, Mar 11 2008

			Triangle T begins:
1;
1, 1;
4, 2, 1;
28, 10, 3, 1;
268, 78, 18, 4, 1;
3164, 798, 156, 28, 5, 1;
43672, 9874, 1714, 268, 40, 6, 1;
682632, 141282, 22368, 3164, 420, 54, 7, 1;
11834536, 2273730, 333910, 43672, 5320, 618, 70, 8, 1;
224283416, 40400466, 5566728, 682632, 77720, 8378, 868, 88, 9, 1; ...
where column k of T = column 0 of T^(k+1)
with column 0 of T = column 0 of T^4 shift right:
column 1 of T = column 0 of T^2;
column 2 of T = column 0 of T^3;
column 3 of T = column 0 of T^4.
Matrix square of T, T^2, begins:
1;
2, 1;
10, 4, 1;
78, 26, 6, 1;
798, 232, 48, 8, 1;
9874, 2578, 486, 76, 10, 1;
141282, 33764, 5888, 864, 110, 12, 1;
2273730, 503910, 82210, 11396, 1390, 150, 14, 1; ...
where column k of T^2 = column 1 of T^(k+1):
column 0 of T^2 = column 1 of T;
column 2 of T^2 = column 1 of T^3;
column 3 of T^2 = column 1 of T^4.
Matrix cube of T, T^3, begins:
1;
3, 1;
18, 6, 1;
156, 48, 9, 1;
1714, 486, 90, 12, 1;
22368, 5888, 1050, 144, 15, 1;
333910, 82210, 14046, 1908, 210, 18, 1;
5566728, 1289928, 211182, 28072, 3120, 288, 21, 1; ...
where column k of T^3 = column 2 of T^(k+1):
column 0 of T^3 = column 2 of T;
column 1 of T^3 = column 2 of T^2;
column 3 of T^3 = column 2 of T^4.
Matrix 4th power of T, T^4, begins:
1;
4, 1;
28, 8, 1;
268, 76, 12, 1;
3164, 864, 144, 16, 1;
43672, 11396, 1908, 232, 20, 1;
682632, 170000, 28072, 3520, 340, 24, 1;
11834536, 2814832, 454848, 57408, 5820, 468, 28, 1; ...
where column k of T^4 = column 3 of T^(k+1):
column 0 of T^4 = column 3 of T = column 0 of T shift left;
column 1 of T^4 = column 3 of T^2;
column 2 of T^4 = column 3 of T^3.

Cf. columns: A138272, A138273, A138274; central terms: A138275; variants: A091351, A094587, A135902.

{T(n, k) = if(k>n||k<0, 0, if(k==n, 1, if(k==0, T(n+2, 3), sum(j=0, n-k, T(n-k, j)*T(j+k-1, k-1))); ); )}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

/* Column k of T = column 0 of T^(k+1): */
{T(n, k) = local(M=if(n==0,Mat(1),matrix(n,n,r,c,if(r>=c,T(r-1,c-1))))); if(k==n, 1, if(k==0, (M^4)[n, 1],(M^(k+1))[n-k+1, 1]))}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

Column k of T^(j+1) = column j of T^(k+1) for j>=0, k>=0.

A370915 A(n, k) = 4^n*Pochhammer(k/4, n). Square array read by ascending antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 5, 2, 1, 0, 45, 12, 3, 1, 0, 585, 120, 21, 4, 1, 0, 9945, 1680, 231, 32, 5, 1, 0, 208845, 30240, 3465, 384, 45, 6, 1, 0, 5221125, 665280, 65835, 6144, 585, 60, 7, 1, 0, 151412625, 17297280, 1514205, 122880, 9945, 840, 77, 8, 1

Offset: 0

Peter Luschny, Mar 06 2024

The sequence of square arrays A(m, n, k) starts: A094587 (m = 1), A370419 (m = 2), A371077(m = 3), this array (m = 4).

			The array starts:
[0] 1,    1,     1,     1,      1,      1,      1,      1,      1, ...
[1] 0,    1,     2,     3,      4,      5,      6,      7,      8, ...
[2] 0,    5,    12,    21,     32,     45,     60,     77,     96, ...
[3] 0,   45,   120,   231,    384,    585,    840,   1155,   1536, ...
[4] 0,  585,  1680,  3465,   6144,   9945,  15120,  21945,  30720, ...
[5] 0, 9945, 30240, 65835, 122880, 208845, 332640, 504735, 737280, ...
.
Seen as the triangle T(n, k) = A(n - k, k):
[0] 1;
[1] 0,      1;
[2] 0,      1,     1;
[3] 0,      5,     2,    1;
[4] 0,     45,    12,    3,   1;
[5] 0,    585,   120,   21,   4,  1;
[6] 0,   9945,  1680,  231,  32,  5, 1;
[7] 0, 208845, 30240, 3465, 384, 45, 6, 1;

Similar square arrays: A094587, A370419, A371077.
Rows: A000012, A001477, A028347, A370914.
Columns: A000007, A007696, A001813, A008545, A047053, A007696, A000407, A034176, A052570 and A034177, A051617, A051618, A051619, A051620.
Cf. A370913 (row sums of triangle), A371026.

A := (n, k) -> 4^n*pochhammer(k/4, n):
for n from 0 to 5 do seq(A(n, k), k = 0..9) od;
T := (n, k) -> A(n - k, k): seq(seq(T(n, k), k = 0..n), n = 0..9);
# Using the exponential generating functions of the columns:
EGFcol := proc(k, len) local egf, ser, n; egf := (1 - 4*x)^(-k/4);
ser := series(egf, x, len+2): seq(n!*coeff(ser, x, n), n = 0..len) end:
seq(lprint(EGFcol(n, 9)), n = 0..5);
# Using the generating polynomials for the rows:
P := (n, x) -> local k; add(Stirling1(n, k)*(-4)^(n - k)*x^k, k=0..n):
seq(lprint([n], seq(P(n, k), k = 0..8)), n = 0..5);
# Implementing the LU decomposition of A:
with(LinearAlgebra):
L := Matrix(7, 7, (n, k) -> A371026(n-1, k-1)):
U := Matrix(7, 7, (n, k) -> binomial(n-1, k-1)):
MatrixMatrixMultiply(L, Transpose(U));

A[n_, k_] := 4^n * Pochhammer[k/4, n]; Table[A[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Mar 06 2024 *)

def A(n, k): return 4**n * rising_factorial(k/4, n)
for n in range(6): print([A(n, k) for k in range(9)])

A(n, k) = 4^n*Product_{j=0..n-1} (j + k/4).
A(n, k) = 4^n*Gamma(k/4 + n) / Gamma(k/4) for k >= 1.
The exponential generating function for column k is (1 - 4*x)^(-k/4). But much more is true: (1 - m*x)^(-k/m) are the exponential generating functions for the columns of the arrays A(m, n, k) = m^n*Pochhammer(k/m, n).
The polynomials P(n, x) = Sum_{k=0..n} Stirling1(n, k)*(-4)^(n-k)*x^k are ordinary generating functions for row n, i.e., A(n, k) = P(n, k).
In A370419 Werner Schulte pointed out how A371025 is related to the LU decomposition of A370419. Here the same procedure can be used and amounts to A = A371026 * transpose(binomial triangle), where '*' denotes matrix multiplication. See the Maple section for an implementation.

A306015 Exponential series expansion of (exp(x*y) + sinh(x) - cosh(x))/(1 - x).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 4, 6, 3, 1, 15, 24, 12, 4, 1, 76, 120, 60, 20, 5, 1, 455, 720, 360, 120, 30, 6, 1, 3186, 5040, 2520, 840, 210, 42, 7, 1, 25487, 40320, 20160, 6720, 1680, 336, 56, 8, 1, 229384, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1

Offset: 0

Peter Luschny, Jun 23 2018

From David Callan, Dec 18 2021: (Start)
For 0 <= k <= n, T(n,k) is the number of nonderangements of size n in which k of the fixed points are colored red. In particular, with D_n the derangement number A000166(n), T(n,0) = n! - D_n. For a general example, T(3,1) = 6 counts the colored permutations R23, R32, 1R3, 3R1, 12R, 21R where the red fixed points are indicated by "R".
For n >= k >= 1, T(n,k) = n!/k!. Proof. In a colored permutation, such as 3R7R516 counted by T(n,k) with n = 7 and k = 2, the R's indicate (red) fixed points and so no information is lost by rank ordering the remaining entries while retaining the placement of the R's: 2R5R314. The result is a permutation of the set consisting of 1,2,...,n-k and k R's; there are n!/k! such permutations and the process is reversible. QED. (End)

			  n |  k = 0       1       2      3      4     5    6   7  8  9
  --+----------------------------------------------------------
  0 |      0
  1 |      1,      1
  2 |      1,      2,      1
  3 |      4,      6,      3,     1
  4 |     15,     24,     12,     4,     1
  5 |     76,    120,     60,    20,     5,    1
  6 |    455,    720,    360,   120,    30,    6,   1
  7 |   3186,   5040,   2520,   840,   210,   42,   7,  1
  8 |  25487,  40320,  20160,  6720,  1680,  336,  56,  8, 1
  9 | 229384, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1

G. C. Greubel, Rows n=0..99 of triangle, flattened

A094587 with an extra first column A002467.

Row sums are A306150.

gf := (exp(x*y) + sinh(x) - cosh(x))/(1 - x):
ser := series(gf, x, 16): L := [seq(n!*coeff(ser, x, n), n=0..14)]:
seq(seq(coeff(L[k+1], y, n), n=0..k), k=0..12);

Join[{0}, With[{nmax = 15}, CoefficientList[CoefficientList[Series[ (Exp[x*y] + Sinh[x] - Cosh[x])/(1 - x), {x, 0, nmax}, {y, 0, nmax}], x], y ]*Range[0, nmax]!] // Flatten ] (* G. C. Greubel, Jul 18 2018 *)

cp's OEIS Frontend

A371077 Square array read by ascending antidiagonals: A(n, k) = 3^n*Pochhammer(k/3, n).

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A112295 Inverse of a double factorial related triangle.

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A162995 A scaled version of triangle A162990.

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A122851 Number triangle T(n,k) = C(k,n-k)*(n-k)!.

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A135902 Triangle T, read by rows, where column k of T = column 0 of T^(k+1) for k>0, with column 0 of T = column 0 of T^3 shift right.

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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.

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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.

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