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Search: id:A137437
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| A137437 |
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Triangular sequence from expansion coefficients of asymptotic Hermite Polynomial from Roman: p(x,t)= (1 + t)^(-x)*Exp[x*(t - t^2/2)]. |
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+0
1
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| 1, 0, 0, 0, -2, 0, 6, 0, -24, 0, 120, 40, 0, -720, -420, 0, 5040, 3948, 0, -40320, -38304, -2240, 0, 362880, 396576, 50400
(list; table; graph; listen)
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OFFSET
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1,5
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COMMENT
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Row sums are:A038205
{1, 0, 0, -2, 6, -24, 160, -1140, 8988, -80864, 809856};
These polynomials are unique in that they connect hyperbolic differential equations, derangements and Hermite orthogonal polynomials;
also the polynomials have a very slow rise in power with n.
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REFERENCES
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Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), page 130
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FORMULA
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p(x,t)= (1 + t)^(-x)*Exp[x*(t - t^2/2)]=Sum[q(x,n)*t^n/n!,{n,0,Infinity}]; out(n,m)=n!*Coefficients(q(x,n)).
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EXAMPLE
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{1},
{0},
{0},
{0, -2},
{0, 6},
{0, -24},
{0, 120, 40},
{0, -720, -420},
{0, 5040, 3948},
{0, -40320, -38304, -2240},
{0, 362880, 396576, 50400}
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MATHEMATICA
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Clear[p, g, m, a]; p[t_] := (1 + t)^(-x)*Exp[x*(t - t^2/2)]; Table[ ExpandAll[n!SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[n!* CoefficientList[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[{{1}, {0}, {0}, {0, -2}, {0, 6}, {0, -24}, {0, 120, 40}, {0, -720, -420}, {0, 5040, 3948}, {0, -40320, -38304, -2240}, {0, 362880, 396576, 50400}}]
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CROSSREFS
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Cf. A038205, A137286.
Sequence in context: A126869 A094233 A094659 this_sequence A021489 A092158 A051831
Adjacent sequences: A137434 A137435 A137436 this_sequence A137438 A137439 A137440
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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 21 2008
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