|
| |
|
|
A060096
|
|
Numerator of coefficients of Euler polynomials (rising powers).
|
|
5
| |
|
|
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; 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
| 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 (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 preceeding formulas). - From W. Lang, Nov 03 2011.
|
|
|
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}]]] (* From Jean-François Alcover, Apr 29 2011 *)
|
|
|
CROSSREFS
| Cf. A060097.
Sequence in context: A025443 A120080 A111700 * A152892 A193002 A181116
Adjacent sequences: A060093 A060094 A060095 * A060097 A060098 A060099
|
|
|
KEYWORD
| sign,easy,tabl,frac
|
|
|
AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Mar 29 2001
|
|
|
EXTENSIONS
| Table rewritten by W. Lang, Oct 31 2011.
|
| |
|
|
