0,4

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)

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

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)

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.

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))

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

Cf. A094587.

Sequence in context: A245958 A138271 A136212 * A136737 A275595 A298906

Adjacent sequences: A136212 A136213 A136214 * A136216 A136217 A136218

nonn,tabl

Paul D. Hanna, Feb 07 2008

approved