A235706 (I + A132440)^3: Coefficients for normalized generalized Laguerre polynomials n!*Lag(n, 3-n, -x).

1, 3, 1, 6, 6, 1, 6, 18, 9, 1, 0, 24, 36, 12, 1, 0, 0, 60, 60, 15, 1, 0, 0, 0, 120, 90, 18, 1, 0, 0, 0, 0, 210, 126, 21, 1, 0, 0, 0, 0, 0, 336, 168, 24, 1, 0, 0, 0, 0, 0, 0, 504, 216, 27, 1, 0, 0, 0, 0, 0, 0, 0, 720, 270, 30, 1

Offset: 0

Author

Tom Copeland, Apr 20 2014

Keywords

• nonn
• tabl
• easy

Comments

The associated Laguerre polynomials n!*Lag(n,3-n,-x) are related to the rook polynomials of a rectangular 3 X n chessboard by R(3,n,x) = n!*x^n*Lag(n,3-n,-1/x), which are also the matching polynomials, or generating function of the number of k-edge matchings, of the complete bipartite graph K(m,n), or biclique (cf. Wikipedia for details).

The formulas here and below can be naturally extended with 3 replaced by any positive integer m. For m = 1 and 2, see unsigned A132013 and A132014. The formulas there can be extrapolated to apply to this matrix.

Examples

			Triangle begins:
  1;
  3,  1;
  6,  6,  1;
  6, 18,  9,  1;
  0, 24, 36, 12,  1;
  0,  0, 60, 60, 15, 1;
  ...
		

Links

• Michel Marcus, Rows n = 0..50 of triangle, flattened

Crossrefs

Cf. A007318, A008306 for a generalization, A132013, A132014, A132440, A238363, A238385.

....................................

With 0th row: 1

n-th row: n!*Lag(n,3-n,-x)

....................................

1st: 1!*Lag(1,2,-x) = A062139(1,k,-x)

2nd: 2!*Lag(2,1,-x) = A105278(2,k,x)

3rd: 3!*Lag(3,0,-x) = A021009(3,k,-x)

4th: 4!*Lag(4,-1,-x) = A111596(4,k,-x)

5th: 5!*Lag(5,-2,-x) = cf. x^2*A062139(3,k,x)

6th: 6!*Lag(6,-3,-x) = cf. x^3*A062137(3,k,-x)

....................................

n-th row: x^(n-3)*3!*Lag(3,n-3,-x)

....................................

1st: x^(-2)*3!Lag(3,-2,-x) = cf. x^(-2)*[x^2*A062139(1,k,x)]

2nd: x^(-1)*3!Lag(3,-1,-x) = x^(-1)*A111596(3,k,-x)

3rd: x^0*3!Lag(3,0,-x) = x^0*A021009(3,k,-x)

4th: x^1*3!Lag(3,1,-x) = x^1*A105278(3,k,x)

5th: x^2*3!Lag(3,2,-x) = x^2*A062139(3,k,-x)

6th: x^3*3!Lag(3,3,-x) = x^3*A062137(3,k,-x)

Programs

Magma

/* As triangle */ [[Binomial(3, n-k)*Factorial(n)/Factorial(k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 28 2017

Mathematica

Table[Binomial[3, n - k] n! / k!, {n, 0, 9}, {k, 0, n}]//Flatten (* Vincenzo Librandi, Jul 28 2017 *)

PARI

T(n,k) = binomial(3,n-k)*n!/k!
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 28 2017

PARI

row(n) = Vecrev(n!*pollaguerre(n, 3-n, -x)); \\ Michel Marcus, Feb 06 2021

Formula

T(n,k) = binomial(3,n-k)*n!/k! = binomial(n,k)*3!/(3-n+k)!.

E.g.f.: exp(y*x)(1+y)^3, so this is an Appell sequence of polynomials with lowering operator L= D= d/dx and raising operator R = x + 3/(1+D).

E.g.f. of inverse matrix is exp(x*y)/(1+y)^3.

Multiply the n-th diagonal of the Pascal matrix A007318 by d(0)=1, d(1)=3, d(2)=6, d(3)=6, and d(n)=0 for n>3 to obtain T.

Row polynomials: n!*Lag(n,3-n,-x) = x^(n-3)*3!*Lag(3,n-3,-x) =

(3!/(3-n)!)*K(-n,3-n+1,-x) where K is Kummer's confluent hypergeometric function (as a limit of n+c as c tends to zero).

T = (I + A132440)^3 = exp[3*(A238385-I)]. I = identity matrix.

Operationally, n!Lag(n,3-n,-:xD:) = x^(n-3)*:Dx:^n*x^(3-n) = x^(-3)*:xD:^n*x^3 = n!*binomial(xD+3,n) = n!*binomial(3,n)*K(-n,3-n+1,-:xD:) where :AB:^n = A^n*B^n for any two operators.

n-th row polynomial: n!*Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*Lag(k,3,-x). - Peter Bala, Jul 25 2021