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A225694
Triangle read by rows of operator ordering coefficients corresponding to the Legendre polynomials L_n(x).
2
1, 1, 1, 7, 10, 7, 17, 103, 103, 17, 203, 2948, 7138, 2948, 203, 583, 20091, 100286, 100286, 20091, 583, 3491, 261462, 2511213, 5092148, 2511213, 261462, 3491, 10481, 1670771, 29075841, 107621147, 107621147, 29075841, 1670771, 10481, 254963
OFFSET
0,4
LINKS
T. Amdeberhan, V. de Angelis, A. Dixit, V. H. Moll and C. Vignat, From sequences to polynomials and back, via operator orderings, 2013.
Carl M. Bender and Gerald V. Dunne, Polynomials and operator orderings, J. Math. Phys. 29 (1988), 1727-1731.
EXAMPLE
Triangle begins:
1
1,1
7,10,7
17,103,103,17
203,2948,7138,2948,203
583,20091,100286,100286,20091,583
...
MAPLE
A225694F := proc(n, k)
add((-1)^(n-k-j)*binomial(n+1, n-k-j)*orthopoly[P](n, I*(j+1/2)), j=0..n-k) ;
%/I^n/n! ;
expand(%) ;
end proc:
A225694 := proc(n, k)
A225694F(n, k) *denom(A225694F(n, 0)) ;
end proc:
seq(seq( A225694(n, k), k=0..n), n=0..10) ; # R. J. Mathar, May 23 2014
MATHEMATICA
F[n_, k_] := F[n, k] = Sum[(-1)^(n - k - j) Binomial[n + 1, n - k - j]* LegendreP[n, I(j + 1/2)], {j, 0, n - k}] /I^n/n!;
T[n_, k_] := F[n, k] LCM @@ Denominator[Table[F[n, j], {j, 0, n}]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 02 2020, after R. J. Mathar *)
CROSSREFS
Sequence in context: A070405 A377931 A010730 * A247191 A317336 A079004
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, May 27 2013
STATUS
approved