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A048998
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Triangle giving coefficients of (n+1)!*B_n(x), where B_n(x) is a Bernoulli polynomial. Rising powers of x.
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10
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1, -1, 2, 1, -6, 6, 0, 12, -36, 24, -4, 0, 120, -240, 120, 0, -120, 0, 1200, -1800, 720, 120, 0, -2520, 0, 12600, -15120, 5040, 0, 6720, 0, -47040, 0, 141120, -141120, 40320, -12096, 0, 241920, 0, -846720, 0, 1693440, -1451520, 362880
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table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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COMMENTS
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See A074909 for generators for the Bernoulli polynomials and connections to the beheaded Pascal triangle and reciprocals of the integers. - Tom Copeland, Nov 17 2014
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REFERENCES
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I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 5th ed., Section 9.62.
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LINKS
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T. D. Noe, Rows n=0..50 of triangle, flattened
Index entries for sequences related to Bernoulli numbers.
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FORMULA
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t*exp(x*t)/(exp(t)-1) = Sum_{n >= 0} B_n(x)*t^n/n!.
a(n,m) = [x^m]((n+1)!*B_n(x)), n>=0, m=0,...,n. - Wolfdieter Lang, Jun 21 2011
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EXAMPLE
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B_0(x)=1; B_1(x)=x-1/2; B_2(x)=x^2-x+1/6; B_3(x)=x^3-3*x^2/2+x/2; B_4(x)=x^4-2*x^3+x^2-1/30; ...
Triangle starts:
1;
-1, 2;
1, -6, 6;
0, 12, -36, 24;
...
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MAPLE
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A048998 := proc(n, k) coeftayl(bernoulli(n, x), x=0, k) ; (n+1)!*% ; end proc:
seq(seq(A048998(n, k), k=0..n), n=0..10) ; # R. J. Mathar, Jun 27 2011
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MATHEMATICA
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Flatten[Table[CoefficientList[(n + 1)! BernoulliB[n, x], x], {n, 0, 10}]] (* T. D. Noe, Jun 21 2011 *)
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CROSSREFS
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Cf. A048999, A074909.
Sequence in context: A095132 A028940 A218853 * A213615 A049019 A133314
Adjacent sequences: A048995 A048996 A048997 * A048999 A049000 A049001
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KEYWORD
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sign,easy,nice,tabl
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Added 'Rising powers of x' in name - Wolfdieter Lang, Jun 21 2011
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STATUS
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approved
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