A278071 - OEIS

%I #27 Sep 03 2023 10:12:36

%S 1,1,-1,6,-4,1,60,-36,9,-1,840,-480,120,-16,1,15120,-8400,2100,-300,

%T 25,-1,332640,-181440,45360,-6720,630,-36,1,8648640,-4656960,1164240,

%U -176400,17640,-1176,49,-1,259459200,-138378240,34594560,-5322240,554400,-40320,2016,-64,1

%N Triangle read by rows, coefficients of the polynomials P(n,x) = (-1)^n*hypergeom( [n,-n], [], x), powers in descending order.

%H H. L. Krall and O. Fink, <a href="https://doi.org/10.1090/S0002-9947-1949-0028473-1">A New Class of Orthogonal Polynomials: The Bessel Polynomials</a>, Trans. Amer. Math. Soc. 65, 100-115, 1949.

%H Herbert E. Salzer, <a href="https://doi.org/10.1090/S0025-5718-1955-0078498-1">Orthogonal Polynomials Arising in the Numerical Evaluation of Inverse Laplace Transforms</a>, Mathematical Tables and Other Aids to Computation, Vol. 9, No. 52 (Oct., 1955), pp. 164-177, (see p.174 and footnote 7).

%F The P(n,x) are orthogonal polynomials. They satisfy the recurrence

%F P(n,x) = ((((4*n-2)*(2*n-3)*x+2)*P(n-1,x)+(2*n-1)*P(n-2,x))/(2*n-3)) for n>=2.

%F In terms of generalized Laguerre polynomials (see the Krall and Fink link):

%F P(n,x) = n!*(-x)^n*LaguerreL(n,-2*n,-1/x).

%e Triangle starts:

%e .       1,

%e .       1,      -1,

%e .       6,      -4,     1,

%e .      60,     -36,     9,    -1,

%e .     840,    -480,   120,   -16,   1,

%e .   15120,   -8400,  2100,  -300,  25,  -1,

%e .  332640, -181440, 45360, -6720, 630, -36, 1,

%e ...

%p p := n -> (-1)^n*hypergeom([n, -n], [], x):

%p ListTools:-Flatten([seq(PolynomialTools:-CoefficientList(simplify(p(n)), x, termorder=reverse), n=0..8)]);

%p # Alternatively the polynomials by recurrence:

%p P := proc(n,x) if n=0 then return 1 fi; if n=1 then return x-1 fi;

%p ((((4*n-2)*(2*n-3)*x+2)*P(n-1,x)+(2*n-1)*P(n-2,x))/(2*n-3));

%p sort(expand(%)) end: for n from 0 to 6 do lprint(P(n,x)) od;

%p # Or by generalized Laguerre polynomials:

%p P := (n,x) -> n!*(-x)^n*LaguerreL(n,-2*n,-1/x):

%p for n from 0 to 6 do simplify(P(n,x)) od;

%t row[n_] := CoefficientList[(-1)^n HypergeometricPFQ[{n, -n}, {}, x], x] // Reverse;

%t Table[row[n], {n, 0, 8}] // Flatten (* _Jean-François Alcover_, Jul 12 2019 *)

%t (* T(n,k)= *) t={};For[n=8,n>-1,n--,For[j=n+1,j>0,j--,PrependTo[t,(-1)^(j-n+1-Mod[n,2])*Product[(2*n-k)*k/(n-k+1),{k,j,n}]]]];t (* _Detlef Meya_, Aug 02 2023 *)

%Y Cf. A278069 (x=1, row sums up to sign), A278070 (x=-1).

%Y T(n,0) = Pochhammer(n, n) (cf. A000407).

%Y T(n,1) = -(n+1)*(2n)!/n! (cf. A002690).

%Y T(n,2) = (n+2)*(2n+1)*(2n-1)!/(n-1)! (cf. A002691).

%Y T(n,n-1) = (-1)^(n+1)*n^2 for n>=1 (cf. A000290).

%Y T(n,n-2) = n^2*(n^2-1)/2 for n>=2 (cf. A083374).

%K sign,tabl

%O 0,4

%A _Peter Luschny_, Nov 10 2016