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Integral form of A137286: Triangle of coefficients of Integral form of recursive orthogonal Hermite polynomials given in Hochstadt's book: n*IP(x, n) = x*P(x, n ) - n*P'(x, n - 2); derived to a constant from the differential recursion: P''(x,n)=x*P'(x,n)-n*P(x,n).
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%I #3 Mar 30 2012 17:34:22

%S 1,-1,1,0,-2,1,5,-2,-3,1,0,18,-5,-4,1,-33,8,42,-9,-5,1,0,-174,33,80,

%T -14,-6,1,279,-48,-555,87,135,-20,-7,1,0,1950,-279,-1380,185,210,-27,

%U -8,1,-2895,384,7920,-975,-2940,345,308,-35,-9,1,0,-25290,2895,24360,-2640,-5628,588,432,-44,-10,1,35685,-3840,-125055

%N Integral form of A137286: Triangle of coefficients of Integral form of recursive orthogonal Hermite polynomials given in Hochstadt's book: n*IP(x, n) = x*P(x, n ) - n*P'(x, n - 2); derived to a constant from the differential recursion: P''(x,n)=x*P'(x,n)-n*P(x,n).

%C Row sums:

%C {1,0, -1, 1, 10, 4, -80, -128, 652, 2104, -5336, -32360}

%D Defined :page 8 and pages 42 - 43 and page 49: Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986

%F The Hermite Integral form is: IH[x,n]=(x*H[x,n]-H'[x,n])/n Which can be done as an integer form: n*IH[x,n]

%e {1},

%e {-1, 1},

%e {0, -2, 1},

%e {5, -2, -3, 1},

%e {0, 18, -5, -4, 1},

%e {-33, 8, 42, -9, -5, 1},

%e {0, -174, 33,80, -14, -6, 1},

%e {279, -48, -555, 87, 135, -20, -7, 1},

%e {0, 1950, -279, -1380, 185, 210, -27, -8, 1},

%e {-2895,384, 7920, -975, -2940, 345, 308, -35, -9, 1},

%e {0, -25290, 2895, 24360, -2640, -5628, 588, 432, -44, -10, 1},

%e {35685, -3840, -125055,12645, 62790, -6090, -9954, 938, 585, -54, -11, 1}

%t P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = x*P[x, n - 1] - n*P[x, n - 2]; DP[x_, n_] := D[P[x, n + 1], x]; Table[ExpandAll[x*P[x, n] - DP[x, n]], {n, 0, 10}]; a = Join[{{1}}, Table[CoefficientList[x*P[x, n] - DP[x, n], x], {n, 0, 10}]]; Flatten[a]

%Y Cf. A137286.

%K uned,tabl,sign

%O 1,5

%A _Roger L. Bagula_, Mar 18 2008