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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.

%I A098503 #34 Feb 07 2018 02:53:07
%S A098503 1,-2,3,4,-20,15,-8,84,-210,105,16,-288,1512,-2520,945,-32,880,-7920,
%T A098503 27720,-34650,10395,64,-2496,34320,-205920,540540,-540540,135135,-128,
%U A098503 6720,-131040,1201200,-5405400,11351340,-9459450,2027025,256,-17408
%N 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.
%H A098503 G. C. Greubel, <a href="/A098503/b098503.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A098503 R. J. Mathar, <a href="http://vixra.org/abs/1303.0013">Gauss-Laguerre and Gauss-Hermite quadrature on 64, 96 and 128 nodes</a>, viXra:1303.0013, Section 3.
%F A098503 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.
%e A098503 2^0 *0! *L(0,1/2,x) = 1.
%e A098503 2^1 *1! *L(1,1/2,x) = -2*x + 3.
%e A098503 2^2 *2! *L(2,1/2,x) = 4*x^2 - 20*x + 15.
%e A098503 2^3 *3! *L(3,1/2,x) = -8*x^3 + 84*x^2 - 210*x + 105.
%e A098503 2^4 *4! *L(4,1/2,x) = 16*x^4 - 288*x^3 + 1512*x^2 - 2520*x + 945.
%e A098503 Triangle begins:
%e A098503     1;
%e A098503    -2,     3;
%e A098503     4,   -20,    15;
%e A098503    -8,    84,  -210,     105;
%e A098503    16,  -288,  1512,   -2520,    945;
%e A098503   -32,   880, -7920,   27720, -34650,   10395;
%e A098503    64, -2496, 34320, -205920, 540540, -540540, 135135;
%t A098503 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 *)
%Y A098503 Columns include (-1)^n times A000079, n/2*A014480. Diagonals include A001147, -A000906, 4*A001881.
%Y A098503 Cf. A223168..A223172 and A223523..A223532.
%K A098503 sign,tabl
%O A098503 0,2
%A A098503 _Ralf Stephan_, Sep 15 2004