A048999 Triangle giving coefficients of (n+1)!*B_n(x), where B_n(x) is a Bernoulli polynomial, ordered by falling powers of x.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• easy
• nice
• tabl

Examples

			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;
...
		

References

• I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 5th ed., Section 9.62.

Links

• T. D. Noe, Rows n = 0..50, flattened
• D. H. Lehmer, A new approach to Bernoulli polynomials, The American mathematical monthly 95.10 (1988): 905-911.
• Index entries for sequences related to Bernoulli numbers.

Crossrefs

Three versions of coefficients of Bernoulli polynomials: A053382/A053383; for reflected version see A196838/A196839; see also A048998 and A048999.

Programs

Mathematica

row[n_] := (n+1)!*Reverse[ CoefficientList[ BernoulliB[n, x], x]]; Flatten[ Table[ row[n], {n, 0, 9}]] (* Jean-François Alcover, Feb 17 2012 *)

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

Formula

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

Extensions

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