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Search: id:A143507
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| A143507 |
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The Hermite polynomials (A060821) with the McMullen's transform substitution:x->x+1/x: p(x,n) = HermiteH[n, x]; q(x,n)=x^(n)*p(x+1/x,n). |
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| 1, 2, 0, 2, 4, 0, 6, 0, 4, 8, 0, 12, 0, 12, 0, 8, 16, 0, 16, 0, 12, 0, 16, 0, 16, 32, 0, 0, 0, -40, 0, -40, 0, 0, 0, 32, 64, 0, -96, 0, -240, 0, -280, 0, -240, 0, -96, 0, 64, 128, 0, -448, 0, -672, 0, -560, 0, -560, 0, -672, 0, -448, 0, 128, 256, 0, -1536, 0, -896, 0, 896, 0, 1680, 0, 896, 0, -896, 0, -1536, 0, 256, 512, 0, -4608, 0
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Row sums are:
{1, 4, 14, 40, 76, -16, -824, -3104, -880, 46144, 200416}.
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FORMULA
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p(x,n) = HermiteH[n, x]; q(x,n)=x^(n)*p(x+1/x,n); t(n,m)=Coefficients(q(x,n)).
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EXAMPLE
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{1},
{2, 0, 2},
{4, 0, 6, 0, 4},
{8, 0, 12, 0, 12, 0, 8},
{16, 0, 16, 0, 12, 0, 16, 0, 16},
{32, 0, 0, 0, -40, 0, -40, 0, 0, 0, 32},
{64, 0, -96, 0, -240, 0, -280, 0, -240, 0, -96, 0, 64},
{128, 0, -448, 0, -672, 0, -560, 0, -560, 0, -672, 0, -448, 0, 128},
{256, 0, -1536, 0, -896,0, 896, 0, 1680, 0, 896, 0, -896, 0, -1536, 0, 256},
{512, 0, -4608, 0, 2304, 0, 10752, 0, 14112, 0, 14112, 0, 10752, 0, 2304, 0, -4608, 0, 512},
{1024, 0, -12800, 0, 23040, 0, 42240, 0, 33600, 0, 26208,0, 33600, 0, 42240, 0, 23040, 0, -12800, 0, 1024}
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MATHEMATICA
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p[x_, n_] = HermiteH[n, x]; Table[FullSimplify[ExpandAll[x^n*p[x + 1/x, n]]], {n, 1, 10}]; Table[CoefficientList[FullSimplify[ExpandAll[x^(n )*p[x + 1/x, n]]], x], {n, 1, 10}]; Flatten[%]
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CROSSREFS
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Cf. A060821.
Sequence in context: A131186 A137320 A137312 this_sequence A071961 A120557 A092594
Adjacent sequences: A143504 A143505 A143506 this_sequence A143508 A143509 A143510
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KEYWORD
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sign,uned,probation
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AUTHOR
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Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Oct 25 2008
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