%I #28 May 07 2020 16:30:40

%S 1,3,-1,15,-5,1,35,-35,7,-1,315,-105,63,-3,1,693,-1155,231,-33,11,-1,

%T 3003,-3003,3003,-143,143,-13,1,6435,-15015,9009,-2145,715,-13,1,-1,

%U 109395,-36465,51051,-7293,12155,-221,17,-17,1,230945,-692835,138567,-46189,46189,-4199,323,-323,19,-1

%N Triangular table of numerators of the coefficients of Laguerre-Sonin polynomials L(1/2,n,x).

%C The corresponding denominator table is given in A130562.

%H G. C. Greubel, <a href="/A131440/b131440.txt">Rows n=0..100 of triangle, flattened</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 775, 22.3.9.

%H W. Lang, <a href="/A130562/a130562.txt">Rational coefficients and more</a>

%F a(n,m) = numerator(L(1/2,n,m)) with L(1/2,n,m) = ((-1)^m)*binomial(n+1/2, n-m)/m!, n>=m>=0, else 0 (taken in lowest terms).

%e Triangle begins:

%e [1];

%e [3,-1];

%e [15,-5,1];

%e [35,-35,7,-1];

%e [315,-105,63,-3,1];

%e [693,-1155,231,-33,11,-1];

%e ...

%e Rationals:

%e [1];

%e [3/2, -1];

%e [15/8, -5/2, 1/2];

%e [35/16, -35/8, 7/4, -1/6];

%e ...

%t T[n_, k_]:= (-1)^k*Binomial[n+1/2, n-k]/k!; Table[Numerator[T[n, k]], {n,0,20}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 14 2018 *)

%o (Python)

%o from sympy import binomial, factorial, Integer

%o def a(n, m): return ((-1)**m * binomial(n + 1/Integer(2), n -m) / factorial(m)).numerator()

%o for n in range(21): print([a(n, m) for m in range(n + 1)]) # _Indranil Ghosh_, Jun 29 2017

%o (PARI) for(n=0,10, for(k=0,n, print1(numerator((-1)^k*binomial(n+1/2, n-k)/k!), ", "))) \\ _G. C. Greubel_, May 14 2018

%K sign,easy,frac,tabl

%O 0,2

%A _Wolfdieter Lang_, Aug 07 2007