A278071 Triangle read by rows, coefficients of the polynomials P(n,x) = (-1)^n*hypergeom( [n,-n], [], x), powers in descending order.

1, 1, -1, 6, -4, 1, 60, -36, 9, -1, 840, -480, 120, -16, 1, 15120, -8400, 2100, -300, 25, -1, 332640, -181440, 45360, -6720, 630, -36, 1, 8648640, -4656960, 1164240, -176400, 17640, -1176, 49, -1, 259459200, -138378240, 34594560, -5322240, 554400, -40320, 2016, -64, 1

Offset: 0

Author

Peter Luschny, Nov 10 2016

Keywords

• sign
• tabl

Examples

			Triangle starts:
.       1,
.       1,      -1,
.       6,      -4,     1,
.      60,     -36,     9,    -1,
.     840,    -480,   120,   -16,   1,
.   15120,   -8400,  2100,  -300,  25,  -1,
.  332640, -181440, 45360, -6720, 630, -36, 1,
...
		

Links

• H. L. Krall and O. Fink, A New Class of Orthogonal Polynomials: The Bessel Polynomials, Trans. Amer. Math. Soc. 65, 100-115, 1949.
• Herbert E. Salzer, Orthogonal Polynomials Arising in the Numerical Evaluation of Inverse Laplace Transforms, Mathematical Tables and Other Aids to Computation, Vol. 9, No. 52 (Oct., 1955), pp. 164-177, (see p.174 and footnote 7).

Crossrefs

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

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

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

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

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

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

Programs

Maple

p := n -> (-1)^n*hypergeom([n, -n], [], x):
ListTools:-Flatten([seq(PolynomialTools:-CoefficientList(simplify(p(n)), x, termorder=reverse), n=0..8)]);
# Alternatively the polynomials by recurrence:
P := proc(n,x) if n=0 then return 1 fi; if n=1 then return x-1 fi;
((((4*n-2)*(2*n-3)*x+2)*P(n-1,x)+(2*n-1)*P(n-2,x))/(2*n-3));
sort(expand(%)) end: for n from 0 to 6 do lprint(P(n,x)) od;
# Or by generalized Laguerre polynomials:
P := (n,x) -> n!*(-x)^n*LaguerreL(n,-2*n,-1/x):
for n from 0 to 6 do simplify(P(n,x)) od;

Mathematica

row[n_] := CoefficientList[(-1)^n HypergeometricPFQ[{n, -n}, {}, x], x] // Reverse;
Table[row[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jul 12 2019 *)
(* 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 *)

Formula

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

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.

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

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