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 A136586 Triangle of coefficients of even modified recursive orthogonal Hermite polynomials given in Hochstadt's book:P(x, n) = x*P(x, n - 1) - n*P(x, n - 2) ;A137286; P2(x,n)=P(x,n)+P(x,n-2). 0

%I #3 Mar 30 2012 17:34:23

%S 0,0,1,-1,0,1,0,-4,0,1,6,0,-8,0,1,0,28,0,-13,0,1,-40,0,78,0,-19,0,1,0,

%T -246,0,171,0,-26,0,1,336,0,-888,0,325,0,-34,0,1,0,2616,0,-2455,0,561,

%U 0,-43,0,1,-3456,0,11670,0,-5745,0,903,0,-53,0,1

%N Triangle of coefficients of even modified recursive orthogonal Hermite polynomials given in Hochstadt's book:P(x, n) = x*P(x, n - 1) - n*P(x, n - 2) ;A137286; P2(x,n)=P(x,n)+P(x,n-2).

%C Row sums are:

%C {0, 1, 0, -3, -1, 16, 20, -100, -260, 680, 3320}

%C The double function Integration is alternating:

%C Table[Integrate[Exp[ -x^2/2]*P2[x, n]*P2[x, m], {x, -Infinity, Infinity}], {n, 0, 10}, {m, 0, 10}];

%C Four Initial conditions were necessary for starting this recursion:

%C P[x, 0] = 1; P[x, 1] = x; P[x, -1] = 0; P[x, -2] = -1;

%F H2(x,n)=A137286(x,n)+A137286(x,n-2)

%e {0},

%e {0, 1},

%e {-1, 0, 1},

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

%e {6, 0, -8, 0, 1},

%e {0, 28, 0, -13, 0, 1},

%e {-40, 0, 78, 0, -19, 0, 1},

%e {0, -246, 0, 171, 0, -26, 0,1},

%e {336, 0, -888, 0, 325, 0, -34, 0, 1},

%e {0, 2616, 0, -2455, 0, 561, 0, -43, 0, 1},

%e {-3456, 0, 11670, 0, -5745, 0, 903, 0, -53, 0, 1}

%t P[x, 0] = 1; P[x, 1] = x; P[x, -1] = 0; P[x, -2] = -1; P[x_, n_] := P[x, n] = x*P[x, n - 1] - n*P[x, n - 2]; P2[x_, n_] := P2[x, n] = P[x, n] + P[x, n - 2]; Table[ExpandAll[P2[x, n]], {n, 0, 10}]; a = Join[{0}, Table[CoefficientList[P2[x, n], x], {n, 0, 10}]]; Flatten[a]

%Y Cf. A137286.

%K uned,tabl,sign

%O 1,8

%A _Roger L. Bagula_, Mar 30 2008

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Last modified October 4 19:37 EDT 2023. Contains 365888 sequences. (Running on oeis4.)