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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 (list; table; graph; refs; listen; history; text; internal format)
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: A266551 A070405 A010730 * A247191 A317336 A079004
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, May 27 2013
STATUS
approved

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Last modified July 18 21:02 EDT 2024. Contains 374388 sequences. (Running on oeis4.)