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A278071
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Triangle read by rows, coefficients of the polynomials P(n,x) = (-1)^n*hypergeom( [n,-n], [], x), powers in descending order.
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2
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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
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OFFSET
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0,4
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LINKS
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FORMULA
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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).
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EXAMPLE
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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,
...
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MAPLE
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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;
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MATHEMATICA
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row[n_] := CoefficientList[(-1)^n HypergeometricPFQ[{n, -n}, {}, x], x] // Reverse;
(* 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 *)
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CROSSREFS
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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).
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KEYWORD
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AUTHOR
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STATUS
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approved
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