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