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A060096 Numerator of coefficients of Euler polynomials (rising powers). 9
1, -1, 1, 0, -1, 1, 1, 0, -3, 1, 0, 1, 0, -2, 1, -1, 0, 5, 0, -5, 1, 0, -3, 0, 5, 0, -3, 1, 17, 0, -21, 0, 35, 0, -7, 1, 0, 17, 0, -28, 0, 14, 0, -4, 1, -31, 0, 153, 0, -63, 0, 21, 0, -9, 1, 0, -155, 0, 255, 0, -126, 0, 30, 0, -5, 1, 691, 0, -1705, 0, 2805, 0, -231, 0, 165, 0, -11, 1, 0, 2073, 0, -3410, 0, 1683, 0, -396 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

From S. Roman, The Umbral Calculus (see the reference in A048854), p. 101, (4.2.10) (corrected): E(n,x)= sum(sum(binomial(n,m)*((-1/2)^j)*j!*S2(n-m,j),j=0..k)*x^m,m=0..n), with S2(n,m)=A008277(n,m) and S2(n,0)=1 if n=0 else 0 (Stirling2).

From Wolfdieter Lang, Oct 31 2011: (Start)

This is the Sheffer triangle (2/(exp(x)+1),x) (which would be called in the above mentioned S. Roman reference Appell for (exp(t)+1)/2) (see p. 27).

  The e.g.f. for the row sums is 2/(1+exp(-x)). The row sums look like A198631(n)/A006519(n+1), n>=0.

  The e.g.f. for the alternating row sums is 2/(exp(x)*(exp(x)+1)). These sums look like (-1)^n*A143074(n)/ A006519(n+1).

  The e.g.f. for the a-sequence of this Sheffer array is 1. The z-sequence has e.g.f. (1-exp(x))/(2*x). This z-sequence is -1/(2*A000027(n))=-1/(2*n) (see the link under A006232 for the definition of a- and z-sequences). This leads to the recurrences given below.

The alternating power sums for the first n positive integers are given by sum((-1)^(n-j)*j^k,j=1..n) = (E(k, x=n+1)+(-1)^n*E(k, x=0))/2, k>=1, n>=1,with the row polynomials E(n, x)(see the Abramowitz-Stegun reference, p. 804, 23.1.4, and an addendum in the W. Lang link under A196837).

(End)

REFERENCES

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.

LINKS

Table of n, a(n) for n=0..85.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

FORMULA

E(n, x)= sum((a(n, m)/b(n, m))*x^m, m=0..n), denominators b(n, m)= A060097(n, m).

From Wolfdieter Lang, Oct 31 2011: (Start)

E.g.f. for E(n, x) is 2*exp(x*z)/(exp(z)+1).

E.g.f. of column no. m, m>=0, is 2*x^{m+1}/(m!*(exp(x)+1)).

Recurrences for E(n,m):=a(n,m)/A060097(n,m) from the Sheffer a-and z-sequence:

E(n,m)=(n/m)*E(n-1,m-1), n>=1,m>=1.

E(n,0)=-n*sum(E(n-1,j)/(2*j),j=0..n-1), n>=1, r(0,0)=1.

(see the Sheffer comments above).

(End)

E(n,m) = binomial(n,m)*sum(((-1)^j)*j!*S2(n-m,j)/2^j ,j=0..n-m), 0<=m<=n, with S2 given by A008277. From S. Roman, The umbral calculus, reference under A048854, eq. (4.2.10), p. 101, with a=1, and a misprint corrected: replace 1/k! by binomial(n,k) (also in the two preceding formulas). - Wolfdieter Lang, Nov 03 2011

The first (m=0) column of the rational triangle is conjectured to be E(n,0) = ((-1)^n)*A198631(n) / A006519(n+1). See also the first column shown in A209308 (different signs). - Wolfdieter Lang, Jun 15 2015

EXAMPLE

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

0:   1

1:  -1    1

2:   0   -1    1

3:   1    0   -3    1

4:   0    1    0   -2    1

5:  -1    0    5    0   -5    1

6:   0   -3    0    5    0   -3    1

7:  17    0  -21    0   35    0   -7    1

8:   0   17    0  -28    0   14    0   -4  1

...

The rational triangle a(n,m)/A060097(n,m) starts

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

0:   1

1: -1/2   1

2:   0   -1    1

3:  1/4   0  -3/2   1

4:   0    1    0   -2    1

5: -1/2   0   5/2   0  -5/2   1

6:   0   -3    0    5    0   -3    1

7: 17/8   0 -21/2   0  35/4   0  -7/2   1

8:   0   17    0  -28    0   14    0   -4  1

...

MAPLE

A060096 := proc(n, m) coeff(euler(n, x), x, m) ; numer(%) ; end proc:

seq(seq(A060096(n, m), m=0..n), n=0..12) ; # R. J. Mathar, Dec 21 2010

MATHEMATICA

Numerator[Flatten[Table[CoefficientList[EulerE[n, x], x], {n, 0, 12}]]] (* Jean-Fran├žois Alcover, Apr 29 2011 *)

CROSSREFS

Cf. A060097.

Sequence in context: A227570 A111700 A233316 * A245756 A152892 A193002

Adjacent sequences:  A060093 A060094 A060095 * A060097 A060098 A060099

KEYWORD

sign,easy,tabl,frac

AUTHOR

Wolfdieter Lang, Mar 29 2001

EXTENSIONS

Table rewritten by Wolfdieter Lang, Oct 31 2011

STATUS

approved

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Last modified November 18 01:16 EST 2017. Contains 294837 sequences.