A098503 Triangle T(n,k) by rows: coefficient [x^(n-k)] of 2^n * n! *L(n,1/2,x), with L the generalized Laguerre polynomials in the Abramowitz-Stegun normalization.

1, -2, 3, 4, -20, 15, -8, 84, -210, 105, 16, -288, 1512, -2520, 945, -32, 880, -7920, 27720, -34650, 10395, 64, -2496, 34320, -205920, 540540, -540540, 135135, -128, 6720, -131040, 1201200, -5405400, 11351340, -9459450, 2027025, 256, -17408

Offset: 0

Author

Ralf Stephan, Sep 15 2004

Keywords

• sign
• tabl

Examples

			2^0 *0! *L(0,1/2,x) = 1.
2^1 *1! *L(1,1/2,x) = -2*x + 3.
2^2 *2! *L(2,1/2,x) = 4*x^2 - 20*x + 15.
2^3 *3! *L(3,1/2,x) = -8*x^3 + 84*x^2 - 210*x + 105.
2^4 *4! *L(4,1/2,x) = 16*x^4 - 288*x^3 + 1512*x^2 - 2520*x + 945.
Triangle begins:
    1;
   -2,     3;
    4,   -20,    15;
   -8,    84,  -210,     105;
   16,  -288,  1512,   -2520,    945;
  -32,   880, -7920,   27720, -34650,   10395;
   64, -2496, 34320, -205920, 540540, -540540, 135135;
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• R. J. Mathar, Gauss-Laguerre and Gauss-Hermite quadrature on 64, 96 and 128 nodes, viXra:1303.0013, Section 3.

Crossrefs

Columns include (-1)^n times A000079, n/2*A014480. Diagonals include A001147, -A000906, 4*A001881.

Cf. A223168..A223172 and A223523..A223532.

Programs

Mathematica

Table[Reverse[Table[2^n*(-1)^k*n!/k!*Binomial[n + 1/2, n - k], {k, 0, n}]], {n, 0, 7}] (* T. D. Noe, Apr 05 2013 *)

Formula

T(n, k) = (-2)^n * (-1)^k * n!/(n-k)! * binomial(n+1/2,k), = (-1)^(n+k) *2^(n-2k) *k! *binomial(2n+1,2k)*binomial(2k,k), n>=0, k<=n.