

A160035


Clausennormalized numerators of the Bernoulli numbers of order 2.


0



1, 0, 1, 0, 3, 0, 5, 0, 7, 0, 45, 0, 7601, 0, 91, 0, 54255, 0, 745739, 0, 3317609, 0, 17944773, 0, 5436374093, 0, 213827575, 0, 641235447783, 0, 249859397004145, 0, 238988952277727
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OFFSET

0,5


COMMENTS

Let B_n{^(k)}(x) denote the Bernoulli polynomials of order k, defined by the generating function
(t/(exp(t)1))^k*exp(x*t) = Sum_{n>=0} B_n{^(k)}(x) t^n/n!
Bernoulli numbers of order 1 (defined as B_n{^(1)}(1)) can be regarded as a pair of sequences B1_n = N1_n / D1_n with
N1_n = A027641, D1_n = A141056 (Clausen).
Similarly Bernoulli numbers of order 2 (defined as B_n{^(2)}(1)) can be regarded as a pair of sequences B2_n = N2_n / D2_n with
N2_n = this sequence, D2_n = A141056 (Clausen).


REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, Boston, Mass., 1974.
C. Jordan, Calculus of Finite Differences, New York, Chelsea, 1965.
N. E. Nørlund, Vorlesungen über Differenzenrechnung, Berlin, SpringerVerlag, 1924.


LINKS

Table of n, a(n) for n=0..32.


EXAMPLE

The Clausennormalized Bernoulli polynomials of order 2 are:
1
2 x  2
6 x^2  12 x + 5
2 x^3  6 x^2 + 5 x  1
30 x^4  120 x^3 + 150 x^2  60 x + 3
2 x^5  10 x^4 + 50/3 x^3  10 x^2 + x + 1/3
42 x^6  252 x^5 + 525 x^4  420 x^3 + 63 x^2 + 42 x  5
The value of these polynomials at x = 1 gives the sequence.


MAPLE

a := proc(n) local g, c, i; g := k > (t/(exp(t)1))^k*exp(x*t): c := proc(n) local i; mul(i, i=select(isprime, map(i>i+1, numtheory[divisors](n)))) end: convert(series(g(2), t, n+8), polynom): seq(i!*c(i)*subs(x=1, coeff(%, t, i)), i=0..n) end:


CROSSREFS

Cf. A120282, A132094, A100615 and A027643.
Sequence in context: A210524 A049283 A141162 * A281648 A210451 A239233
Adjacent sequences: A160032 A160033 A160034 * A160036 A160037 A160038


KEYWORD

sign


AUTHOR

Peter Luschny, Apr 30 2009


STATUS

approved



