A225694 Triangle read by rows of operator ordering coefficients corresponding to the Legendre polynomials L_n(x).

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

Table of n, a(n) for n=0..36.

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

Adjacent sequences: A225691 A225692 A225693 * A225695 A225696 A225697

Keyword

nonn,tabl

Author

N. J. A. Sloane, May 27 2013

Status

approved