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A122753
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Triangle of coefficients of (1 - x)^n*B_n(x/(1 - x)), where B_n(x) is the n-th Bell polynomial.
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14
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1, 0, 1, 0, 1, 0, 1, 1, -1, 0, 1, 4, -5, 1, 0, 1, 11, -14, 1, 2, 0, 1, 26, -24, -29, 36, -9, 0, 1, 57, 1, -244, 281, -104, 9, 0, 1, 120, 225, -1259, 1401, -454, -83, 50, 0, 1, 247, 1268, -5081, 4621, 911, -3422, 1723, -267, 0, 1, 502, 5278, -16981, 5335, 30871
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OFFSET
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0,12
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COMMENTS
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The n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A048993(n,j)*x^j*(1 - x)^(n - j), where A048993 is the triangle of Stirling numbers of second kind.
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 824-825.
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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FORMULA
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E.g.f.: exp((x/(1 - x))*(exp((1 - x)*y) - 1).
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EXAMPLE
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Triangle begins:
1;
0, 1;
0, 1;
0, 1, 1, -1;
0, 1, 4, -5, 1;
0, 1, 11, -14, 1, 2;
0, 1, 26, -24, -29, 36, -9;
0, 1, 57, 1, -244, 281, -104, 9;
0, 1, 120, 225, -1259, 1401, -454, -83, 50;
0, 1, 247, 1268, -5081, 4621, 911, -3422, 1723, -267;
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MATHEMATICA
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Table[CoefficientList[Sum[StirlingS2[m, n]*x^n*(1-x)^(m-n), {n, 0, m}], x], {m, 0, 10}]//Flatten
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PROG
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(Maxima)
P(x, n) := expand(sum(stirling2(n, j)*x^j*(1 - x)^(n - j), j, 0, n))$
T(n, k) := ratcoef(P(x, n), x, k)$
(Sage)
def p(n, x): return sum( stirling_number2(n, j)*x^j*(1-x)^(n-j) for j in (0..n) )
def T(n): return ( p(n, x) ).full_simplify().coefficients(sparse=False)
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CROSSREFS
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KEYWORD
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sign,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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