login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 4
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

LINKS

Table of n, a(n) for n=0..44.

Wikipedia, Appell sequence

Wikipedia, Sheffer sequence

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)

EXAMPLE

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; ...

MATHEMATICA

(* The function RiordanArray is defined in A256893. *)

RiordanArray[1/(1 - 3 #)^(2/3)&, #&, 9, True] // Flatten (* Jean-Fran├žois Alcover, Jul 19 2019 *)

PROG

(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(""))

CROSSREFS

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

Cf. A094587.

Sequence in context: A110682 A110327 A105615 * A121334 A126450 A235608

Adjacent sequences:  A136213 A136214 A136215 * A136217 A136218 A136219

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Feb 07 2008

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 21 01:18 EDT 2019. Contains 328291 sequences. (Running on oeis4.)