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A136586
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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).
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0
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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, -246, 0, 171, 0, -26, 0, 1, 336, 0, -888, 0, 325, 0, -34, 0, 1, 0, 2616, 0, -2455, 0, 561, 0, -43, 0, 1, -3456, 0, 11670, 0, -5745, 0, 903, 0, -53, 0, 1
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OFFSET
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1,8
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COMMENTS
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Row sums are:
{0, 1, 0, -3, -1, 16, 20, -100, -260, 680, 3320}
The double function Integration is alternating:
Table[Integrate[Exp[ -x^2/2]*P2[x, n]*P2[x, m], {x, -Infinity, Infinity}], {n, 0, 10}, {m, 0, 10}];
Four Initial conditions were necessary for starting this recursion:
P[x, 0] = 1; P[x, 1] = x; P[x, -1] = 0; P[x, -2] = -1;
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LINKS
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Table of n, a(n) for n=1..66.
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FORMULA
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H2(x,n)=A137286(x,n)+A137286(x,n-2)
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EXAMPLE
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{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, -246, 0, 171, 0, -26, 0,1},
{336, 0, -888, 0, 325, 0, -34, 0, 1},
{0, 2616, 0, -2455, 0, 561, 0, -43, 0, 1},
{-3456, 0, 11670, 0, -5745, 0, 903, 0, -53, 0, 1}
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MATHEMATICA
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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]
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CROSSREFS
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Cf. A137286.
Sequence in context: A147308 A110064 A021253 * A092746 A097898 A154884
Adjacent sequences: A136583 A136584 A136585 * A136587 A136588 A136589
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KEYWORD
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uned,tabl,sign
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AUTHOR
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Roger L. Bagula, Mar 30 2008
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STATUS
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approved
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