A343861 Coefficient triangle of generalized Laguerre polynomials n!*L(n,n,x) (rising powers of x).

1, 2, -1, 12, -8, 1, 120, -90, 18, -1, 1680, -1344, 336, -32, 1, 30240, -25200, 7200, -900, 50, -1, 665280, -570240, 178200, -26400, 1980, -72, 1, 17297280, -15135120, 5045040, -840840, 76440, -3822, 98, -1, 518918400, -461260800, 161441280, -29352960, 3057600, -188160, 6720, -128, 1

Offset

0,2

Links

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

Wikipedia, Laguerre polynomials

Index entries for sequences related to Laguerre polynomials

Formula

T(n, k) = (-1)^k * n! * binomial(2*n,n-k)/k! = (-1)^k * (2*n)! * binomial(n,k)/(k+n)!.

T(n, 0) = A001813(n).

T(n, 1) = -A092956(n-1).

Sum_{k=0..n} T(n, k) = A006902(n).

Sum_{k=0..n} (-1)^k * T(n, k) = A082545(n).

Example

The triangle begins:
       1;
       2,      -1;
      12,      -8,      1;
     120,     -90,     18,     -1;
    1680,   -1344,    336,    -32,    1;
   30240,  -25200,   7200,   -900,   50,  -1;
  665280, -570240, 178200, -26400, 1980, -72, 1;

Mathematica

T[n_, k_] := (-1)^k * (2*n)! * Binomial[n, k]/(k + n)!; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Amiram Eldar, May 11 2021 *)

Prog

(PARI) T(n, k) = (-1)^k*(2*n)!*binomial(n, k)/(k+n)!;
(PARI) row(n) = Vecrev(n!*pollaguerre(n, n));
(Magma) [(-1)^k*Factorial(n-k)*Binomial(n, k)*Binomial(2*n, n+k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 11 2022
(SageMath)
def A343861(n, k): return (-1)^k*factorial(n-k)*binomial(n, k)*binomial(2*n, n+k)
flatten([[A343861(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 11 2022

Crossrefs

For k=0..1 the (unsigned) columns give A001813, A092956(n-1).

Row sums (signed) give A006902, row sums (unsigned) give A082545.

Cf. A066667, A062137, A062138, A062139, A062140.

Sequence in context: A074966 A128413 A058843 * A130559 A135256 A090586

Adjacent sequences: A343858 A343859 A343860 * A343862 A343863 A343864

Keyword

sign,tabl

Author

Seiichi Manyama, May 01 2021

Status

approved