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A123235
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Strange recursive polynomial triangular array related to the Bessel function has two levels for each power of the polynomial variable: p(k, x) = 2*(k - 1)*p(k - 1, x) -x*p(k - 2, x).
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1
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1, 1, 1, 2, 1, 8, 3, -1, 48, 16, -7, 384, 120, -59, 1, 3840, 1152, -606, 17, 46080, 13440, -7392, 263, -1, 645120, 184320, -104640, 4288, -31, 10321920, 2903040, -1687680, 76000, -759, 1, 185794560, 51609600, -30562560, 1472640, -17950, 49
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| The Bessel recursive polynomial from Jahnke and Emde is: Z(p-1,x)+Z(p+1,x)=(2*p/x)*Z(p,x) Rearranging gives: x*Z(p+1,x)=2*p*Z(p,x)-x*Z(p-1,x) Replace p with k-1: x*Z(k,x)=2*(k-1)*Z(k-1,x)-x*Z(k-2,x) By doing it in two levels it appears that a Bessel polynomial results. Triangle row sums give A093856
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REFERENCES
| Eugene Jahnke and Fritz Emde, Table of Functions with Formulae and Curves, Dover Book, New York,1945, page 144
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FORMULA
| p(k, x) = 2*(k - 1)*p(k - 1, x) - x*p(k - 2, x)
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EXAMPLE
| {1},
{1, 1},
{2, 1},
{8, 3, -1},
{48, 16, -7},
{384, 120, -59, 1},
{3840, 1152, -606, 17},
{46080, 13440, -7392, 263, -1}
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MATHEMATICA
| p[0, x] = 1; p[1, x] = x + 1; p[k_, x_] := p[k, x] = 2*(k - 1)*p[k - 1, x] - x*p[k - 2, x]; w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Floor[w]
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CROSSREFS
| Cf. A093856.
Sequence in context: A011208 A001281 A065826 * A176052 A140273 A021462
Adjacent sequences: A123232 A123233 A123234 * A123236 A123237 A123238
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KEYWORD
| uned,tabl,sign
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AUTHOR
| Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 06 2006
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