A343890 Coefficient triangle of generalized Laguerre polynomials n!*L(n,n+1,x) (rising powers of x).

1, 3, -1, 20, -10, 1, 210, -126, 21, -1, 3024, -2016, 432, -36, 1, 55440, -39600, 9900, -1100, 55, -1, 1235520, -926640, 257400, -34320, 2340, -78, 1, 32432400, -25225200, 7567560, -1146600, 95550, -4410, 105, -1, 980179200, -784143360, 249500160, -41583360, 3998400, -228480, 7616, -136, 1

Offset

0,2

Links

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

Wikipedia, Laguerre polynomials

Index entries for sequences related to Laguerre polynomials

Formula

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

Example

The triangle begins:
        1;
        3,      -1;
       20,     -10,      1;
      210,    -126,     21,     -1;
     3024,   -2016,    432,    -36,    1;
    55440,  -39600,   9900,  -1100,   55,  -1;
  1235520, -926640, 257400, -34320, 2340, -78, 1;

Mathematica

T[n_, k_] := (-1)^k * (2*n + 1)! * Binomial[n, k]/(k + n + 1)!; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Amiram Eldar, May 03 2021 *)

Prog

(PARI) T(n, k) = (-1)^k*(2*n+1)!*binomial(n, k)/(k+n+1)!;
(PARI) row(n) = Vecrev(n!*pollaguerre(n, n+1));

Crossrefs

Row sums (signed) give A343896, row sums (unsigned) give A343832.

Cf. A343861.

Sequence in context: A002380 A274075 A038455 * A067802 A181832 A139723

Adjacent sequences: A343887 A343888 A343889 * A343891 A343892 A343893

Keyword

sign,tabl

Author

Seiichi Manyama, May 03 2021

Status

approved