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A048999
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Triangle giving coefficients of (n+1)!*B_n(x), where B_n(x) is a Bernoulli polynomial, ordered by falling powers of x.
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7
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1, 2, -1, 6, -6, 1, 24, -36, 12, 0, 120, -240, 120, 0, -4, 720, -1800, 1200, 0, -120, 0, 5040, -15120, 12600, 0, -2520, 0, 120, 40320, -141120, 141120, 0, -47040, 0, 6720, 0, 362880, -1451520, 1693440, 0, -846720, 0, 241920, 0, -12096, 3628800
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OFFSET
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0,2
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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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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^(n-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=1 => a(0) = 1;
B_1(x)=x-1/2 => a(1..2) = 2, -1;
B_2(x)=x^2-x+1/6 => a(3..5) = 6, -6, 1;
B_3(x)=x^3-3*x^2/2+x/2 => a(6..9) = 24, -36, 12, 0;
B_4(x)=x^4-2*x^3+x^2-1/30 => a(10..14) = 120, -240, 120, 0, -4;
...
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MATHEMATICA
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row[n_] := (n+1)!*Reverse[ CoefficientList[ BernoulliB[n, x], x]]; Flatten[ Table[ row[n], {n, 0, 9}]] (* Jean-François Alcover, Feb 17 2012 *)
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PROG
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(PARI) P=Pol(t*exp(x*t)/(exp(t)-1)); for(i=0, 15, z=polcoeff(P, i, t)*i!; print(z" => ", (i+1)!*Vec(z))) /* print B_n's and list of normalized coefficients */ \\ M. F. Hasler, Jun 21 2011
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Name clarified by adding 'Falling powers of x.' from Wolfdieter Lang, Jun 21 2011
Values corrected by inserting a(9),a(20),a(35)=0 by M. F. Hasler, Jun 21 2011
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STATUS
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approved
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