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Search: id:A137337
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| A137337 |
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A polynomial expansion like those used in the Domb expansion in Hill's Statistical Mechanics: n!*G.f of t*Exp[ -2*t*(1 - 2x)] and recursive polynomial derived as: p(x, n) = 4 (-1 + 2 x)*p(x, n - 1) - 4 (1 - 4 x + 4 x^2)*p(x, n - 2). |
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+0
1
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| 0, 1, -4, 8, 12, -48, 48, -32, 192, -384, 256, 80, -640, 1920, -2560, 1280, -192, 1920, -7680, 15360, -15360, 6144, 448, -5376, 26880, -71680, 107520, -86016, 28672, -1024, 14336, -86016, 286720, -573440, 688128, -458752, 131072, 2304, -36864, 258048, -1032192, 2580480, -4128768, 4128768
(list; table; graph; listen)
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OFFSET
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1,3
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COMMENT
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Row sums:
{0, 1, 4, 12, 32, 80, 192, 448, 1024, 2304, 5120};
These polynomials are related to Bernoulli polynomial expansions
as to kind
and Domb statistical mechanics polynomials as to method of formation.
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REFERENCES
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Terrell Hill, Statistical Mechanics, Dover, 1987, page 333 ff
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FORMULA
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n!*G.f of t*Exp[ -2*t*(1 - 2x)] and recursive polynomial derived as: p(x, n) = 4 (-1 + 2 x)*p(x, n - 1) - 4 (1 - 4 x + 4 x^2)*p(x, n - 2)
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EXAMPLE
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{0},
{1},
{-4, 8},
{12, -48, 48},
{-32, 192, -384,256},
{80, -640, 1920, -2560, 1280},
{-192,1920, -7680, 15360, -15360, 6144},
{448, -5376, 26880, -71680, 107520, -86016, 28672},
{-1024, 14336, -86016,286720, -573440, 688128, -458752, 131072},
{2304, -36864, 258048,-1032192, 2580480, -4128768, 4128768, -2359296, 589824}, {-5120, 92160, -737280, 3440640, -10321920, 20643840, -27525120, 23592960,-11796480, 2621440}
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MATHEMATICA
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Clear[p] p[t_] = t*Exp[ -2*t*(1 - 2x)]; g = Table[ ExpandAll[n!*SeriesCoefficient[Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Join[{{0}}, Table[ CoefficientList[ExpandAll[n!*SeriesCoefficient[Series[p[t], {t, 0, 30}], n]], x], {n, 0, 10}]]; Flatten[a] (*Polynomial recursion*) Clear[p] p[x, 0] = 0; p[x, 1] = 1; p[x, 2] = -4 + 8*x; p[x_, n_] := p[x, n] = 4 (-1 + 2 x)*p[x, n - 1] - 4 (1 - 4 x + 4 x^2)*p[x, n - 2]; Table[ExpandAll[p[x, n]], {n, 0, Length[g] - 1}]
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CROSSREFS
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Adjacent sequences: A137334 A137335 A137336 this_sequence A137338 A137339 A137340
Sequence in context: A050908 A038563 A083492 this_sequence A061517 A058759 A032474
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KEYWORD
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uned,tabl,sign
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 07 2008
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