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A235706
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(I + A132440)^3: Coefficients for normalized generalized Laguerre polynomials n!*Lag(n, 3-n, -x).
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7
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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
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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).
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
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EXAMPLE
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Triangle begins:
1;
3, 1;
6, 6, 1;
6, 18, 9, 1;
0, 24, 36, 12, 1;
0, 0, 60, 60, 15, 1;
...
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MATHEMATICA
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Table[Binomial[3, n - k] n! / k!, {n, 0, 9}, {k, 0, n}]//Flatten (* Vincenzo Librandi, Jul 28 2017 *)
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PROG
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(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
(Magma) /* As triangle */ [[Binomial(3, n-k)*Factorial(n)/Factorial(k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 28 2017
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CROSSREFS
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....................................
With 0th row: 1
n-th row: n!*Lag(n,3-n,-x)
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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)
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n-th row: x^(n-3)*3!*Lag(3,n-3,-x)
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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)
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KEYWORD
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AUTHOR
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STATUS
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approved
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