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Search: id:A053382
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| A053382 |
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Triangle T(n,k) giving numerator of coefficient of x^(n-k) in Bernoulli polynomial B(n, x), n >= 0, 0<=k<=n. |
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+0
2
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| 1, 1, -1, 1, -1, 1, 1, -3, 1, 0, 1, -2, 1, 0, -1, 1, -5, 5, 0, -1, 0, 1, -3, 5, 0, -1, 0, 1, 1, -7, 7, 0, -7, 0, 1, 0, 1, -4, 14, 0, -7, 0, 2, 0, -1, 1, -9, 6, 0, -21, 0, 2, 0, -3, 0, 1, -5, 15, 0, -7, 0, 5, 0, -3, 0, 5, 1, -11, 55, 0, -11, 0, 11, 0, -11, 0, 5, 0, 1, -6, 11, 0, -33, 0, 22, 0
(list; table; graph; listen)
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OFFSET
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0,8
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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, 1964 (and various reprintings), p. 809.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 48, [14a].
H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.
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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, December 1972 [alternative scanned copy].
H. Pan and Z. W. Sun, New identities involving Bernoulli and Euler polynomials
Index entries for sequences related to Bernoulli numbers.
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FORMULA
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B(m, x) = Sum{n=0..m, 1/(n+1)*Sum[k=0..n, (-1)^k*C(n, k)*(x+k)^m ]].
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EXAMPLE
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The polynomials B(0,x), B(1,x), B(2,x), ... are 1; x-1/2; x^2-x+1/6; x^3-3/2*x^2+1/2*x; x^4-2*x^3+x^2-1/30; x^5-5/2*x^4+5/3*x^3-1/6*x; x^6-3*x^5+5/2*x^4-1/2*x^2+1/42; ...
1, -1/2, 1, 1/6, -1, 1, 0, 1/2, -3/2, 1, -1/30, 0, 1, -2, 1, 0, -1/6, 0, 5/3, -5/2, 1, 1/42, 0, -1/2, 0, 5/2, -3, 1, ... = A053382/A053383 (reflected)
1, 1, -1/2, 1, -1, 1/6, 1, -3/2, 1/2, 0, 1, -2, 1, 0, -1/30, 1, -5/2, 5/3, 0, -1/6, 0, 1, -3, 5/2, 0, -1/2, 0, 1/42, ... = A053382/A053383
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MAPLE
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with(numtheory); bernoulli(n, x);
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CROSSREFS
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Cf. A053383, A048998, A048999.
Sequence in context: A051834 A062719 A117417 this_sequence A031253 A122779 A120323
Adjacent sequences: A053379 A053380 A053381 this_sequence A053383 A053384 A053385
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KEYWORD
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sign,easy,nice,frac,tabl
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AUTHOR
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njas, Jan 06 2000
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EXTENSIONS
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More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jan 10 2000
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