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Search: id:A137452
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| A137452 |
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A triangular sequence from the coefficients an Abel polynomial given as an example in Roman:a0=1; a(x,n)=x*(x-a0**n)^(n-1). |
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1
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| 1, 0, 1, 0, -2, 1, 0, 9, -6, 1, 0, -64, 48, -12, 1, 0, 625, -500, 150, -20, 1, 0, -7776, 6480, -2160, 360, -30, 1, 0, 117649, -100842, 36015, -6860, 735, -42, 1, 0, -2097152, 1835008, -688128, 143360, -17920, 1344, -56, 1, 0, 43046721, -38263752, 14880348, -3306744, 459270, -40824, 2268, -72, 1, 0
(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:
{1, 1, -1, 4, -27, 256, -3125, 46656, -823543, 16777216, -387420489}
The Abel Polynomials are associated with the Abel operator:t*Exp[y*t]*p(x)=t*p(x+y).
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REFERENCES
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Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 14 and 29
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FORMULA
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a(x,n)=x*(x-a0*n)^(n-1)
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EXAMPLE
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{1},
{0, 1},
{0, -2, 1},
{0, 9, -6, 1},
{0, -64, 48, -12, 1},
{0, 625, -500, 150, -20, 1},
{0, -7776, 6480, -2160, 360, -30, 1},
{0, 117649, -100842, 36015, -6860, 735, -42, 1},
{0, -2097152, 1835008, -688128, 143360, -17920, 1344, -56, 1},
{0, 43046721, -38263752, 14880348, -3306744, 459270, -40824, 2268, -72,1},
{0, -1000000000, 900000000, -360000000, 84000000, -12600000, 1260000, -84000, 3600, -90, 1}
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MATHEMATICA
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Clear[A, x, a] a0 = 1 a[x, 0] = 1; a[x, 1] = x; a[x_, n_] := x*(x - a0*n)^(n - 1); Table[Expand[a[x, n]], {n, 0, 10}]; a1 = Table[CoefficientList[a[x, n], x], {n, 0, 10}]; Flatten[a1]
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CROSSREFS
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Adjacent sequences: A137449 A137450 A137451 this_sequence A137453 A137454 A137455
Sequence in context: A076341 A110510 A051122 this_sequence A111595 A021478 A115563
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KEYWORD
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tabl,uned,sign
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 18 2008
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