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A196838 Numerators of coefficients of Bernoulli polynomials with rising powers of the variable. 20
1, -1, 1, 1, -1, 1, 0, 1, -3, 1, -1, 0, 1, -2, 1, 0, -1, 0, 5, -5, 1, 1, 0, -1, 0, 5, -3, 1, 0, 1, 0, -7, 0, 7, -7, 1, -1, 0, 2, 0, -7, 0, 14, -4, 1, 0, -3, 0, 2, 0, -21, 0, 6, -9, 1, 5, 0, -3, 0, 5, 0, -7, 0, 15, -5, 1, 0, 5, 0, -11, 0, 11, 0, -11, 0, 55, -11, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

The denominator triangle is found under A196839.

This is the row reversed triangle A053382.

From Wolfdieter Lang, Oct 25 2011: (Start)

This is the Sheffer triangle (z/(exp(z)-1),z), meaning that the column e.g.f.'s are as given below in the formula section. In Roman's book `The Umbral Calculus`, Ch. 2, 5., p. 26ff this is called Appell for (exp(t)-1)/t (see A048854 for the reference).

The e.g.f. for the a- and z-sequence for this Sheffer triangle is 1 and (x-exp(x)+1)/x^2, respectively. See the link under A006232 for the definition. The z-sequence is z(n) = -1/(2*A000217(n+1)). This leads to the recurrence relations given below.

The e.g.f. for the row sums is x/(1-exp(-x)), leading to the rational sequence A164555(n)/A027664(n). The e.g.f. of the alternating row sums is

  x/(exp(x)*(exp(x)-1)), leading to the rational sequence

  (-1)^n*A164558(n)/A027664(n).

(End)

REFERENCES

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1991 (Seventh printing).Second ed. 1994.

See also A053382.

LINKS

Table of n, a(n) for n=0..77.

Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.

D. H. Lehmer, A new approach to Bernoulli polynomials, The American mathematical monthly 95.10 (1988): 905-911.

See A053382.

FORMULA

a(n,m) = numerator([x^m]Bernoulli(n,x)), n>=0, m=0..n.

E.g.f. of Bernoulli(n,x): z*exp(x*z)/(exp(z)-1).

See the Graham et al. reference, eq. (7.80), p. 354.

From Wolfdieter Lang, Oct 25 2011: (Start)

The e.g.f. for column no. m>=0 of the rational triangle B(n,m):=a(n,m)/A096839(n,m) is x^(m+1)/(m!*(exp(x)-1)).

(see the Sheffer-Appell comment above).

The Sheffer a-sequence, given as comment above, leads to the recurrence r(n,m)=(n/m)*r(n-1,m-1), n>=1, m>=1. E.g., -1/6 = B(5,1) = (5/1)*B(4,0)= -5/30 = -1/6.

The Sheffer z-sequence, given as comment above, leads to the recurrence

B(n,0) = n*sum(z(j)*B(n-1,j),j=0..n-1), n>=1. B(0,0)=1.

E.g., -1/30 = B(4,0) = 4*((-1/2)*0 + (-1/6)*(1/2) + (-1/12)*(-3/2) + (-1/20)*1) = -1/30.

(End)

EXAMPLE

The triangle starts with

n\m 0  1  2  3  4  5  6  7  8 ...

0:  1

1: -1  1

2:  1 -1  1

3:  0  1 -3  1

4: -1  0  1 -2  1

5:  0 -1  0  5 -5  1

6:  1  0 -1  0  5 -3  1

7:  0  1  0 -7  0  7 -7  1

8: -1  0  2  0 -7  0 14 -4  1

...

The rational triangle a(n,m)/A196839(n,m) starts with:

n\m   0     1     2    3    4    5     6    7   8 ...

0:    1

1:  -1/2    1

2:   1/6   -1     1

3:    0    1/2  -3/2   1

4:  -1/30   0     1   -2    1

5:    0   -1/6    0   5/3 -5/2   1

6:   1/42   0   -1/2   0   5/2  -3     1

7:    0    1/6    0  -7/6   0   7/2  -7/2   1

8:  -1/30   0    2/3   0  -7/3   0   14/3  -4   1

...

E.g., Bernoulli(2,x) = (1/6)*x^0 - 1*x^1 + 1*x^2.

CROSSREFS

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

Sequence in context: A261609 A228488 A062172 * A284376 A088205 A244657

Adjacent sequences:  A196835 A196836 A196837 * A196839 A196840 A196841

KEYWORD

sign,easy,tabl,frac

AUTHOR

Wolfdieter Lang, Oct 23 2011

STATUS

approved

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Last modified February 23 13:38 EST 2018. Contains 299581 sequences. (Running on oeis4.)