|
COMMENT
|
In the following the expression [n odd] is 1 if n is odd, 0 otherwise.
(+) W_n(0) = E_n are the Euler (or secant) numbers A122045.
(+) W_n(1) = T_n are the signed tangent numbers, see A009006.
(+) W_{n-1}(1) n / (4^n - 2^n) = B_n gives for n > 1 the Bernoulli number A027641/A027642.
(+) W_n(-1) 2^{-n}(n+1) = G_n the Genocchi number A036968.
(+) W_n(1/2) 2^{n} are the signed generalized Euler (Springer) number, see A001586.
(+) | W_n([n odd]) | the number of alternating permutations A000111.
(+) | W_n([n odd]) / n! | for 0<=n the Euler zeta number A099612/A099617 (see Wikipedia on Bernoulli number).
Contribution from Peter Bala (pbala(AT)talktalk.net), Jun 10 2009: (Start)
The Swiss-Knife polynomials W_n(x) may be expressed in terms of the Bernoulli polynomials B(n,x) as
... W_n(x) = 4^(n+1)/(2*n+2)*[B(n+1,(x+3)/4) - B(n+1,(x+1)/4)].
The Swiss_Knife polynomials are, apart from a multiplying factor, examples of generalised Bernoulli polynomials.
Let X be the Dirichlet character modulus 4 defined by X(4*n+1) = 1, X(4*n+3) = -1 and X(2*n) = 0. The generalised Bernoulli polynomials B(X;n,x), n = 1,2,..., associated with the character X are defined by means of the generating function
... t*exp(x*t)*(exp(t)-exp(3*t))/(exp(4*t)-1) = sum {n = 1..inf} B(X;n,x)*t^n/n!.
The first few values are B(X;1,x) = -1/2, B(X;2,x) = -x, B(X,3,x) = -3/2*(x^2-1) and B(X;4,x) = -2*(x^3-3*x).
In general, W_n(x) = -2/(n+1)*B(X;n+1,x).
For the theory of generalised Bernoulli polynomials associated to a periodic arithmetical function see [Cohen, Section 9.4].
The generalised Bernoulli polynomials may be used to evaluate twisted sums of k-th powers. For the present case the result is
sum{n = 0..4*N-1} X(n)*n^k = 1^k - 3^k + 5^k - 7^k + ... - (4*N-1)^k
= [B(X;k+1,4*N) - B(X;k+1,0)]/(k+1) = [W_k(0) - W_k(4*N)]/2.
For the proof apply [Cohen, Corollary 9.4.17 with m = 4 and x = 0].
The generalised Bernoulli polynomials and the Swiss-Knife polynomials are also related to infinite sums of powers through their Fourier series - see the formula section below. For a table of the coefficients of generalised Bernoulli polynomials attached to a Dirichlet character modulus 8 see A151751.
(End)
Contribution from Peter Luschny (peter(AT)luschny.de), Jul 12 2009: (Start)
The Swiss-Knife polynomials provide a general formula for alternating sums of powers similar to the formula which are provided by the Bernoulli polynomials for non-alternating sums of powers (see the Luschny link). Sequences covered by this formula include A001057, A062393, A062392, A011934, A144129, A077221, A137501, A046092. (End)
Contribution from Peter Luschny (peter(AT)luschny.de), Jul 07 2009: (Start)
W_n(k), k=0,1,...
W_0: 1, 1, 1, 1, 1, 1, .......... A000012
W_1: 0, 1, 2, 3, 4, 5, .......... A001477
W_2: -1, 0, 3, 8, 15, 24, ....... A067998
W_3: 0, -2, 2, 18, 52, 110, ..... A136434
W_4: 5, 0, -3, 32, 165, 480, ....
W_n(k), n=0,1,...
k=0: 1, 0, -1, 0, 5, 0, -61, ..... A122045
k=1: 1, 1, 0, -2, 0, 16, 0, ...... A155585
k=2: 1, 2, 3, 2, -3, 2, 63, ...... A119880
k=3: 1, 3, 8, 18, 32, 48, 128, ... A119881
k=4: 1, 4, 15, 52, 165, 484, .....
(End)
|
|
MAPLE
|
w := proc(n, x) local v, k, pow, chen; pow := (a, b) -> if a = 0 and b = 0 then 1 else a^b fi; chen := proc(m) if irem(m+1, 4) = 0 then RETURN(0) fi; 1/((-1)^iquo(m+1, 4) *2^iquo(m, 2)) end; add(add((-1)^v*binomial(k, v)*pow(v+x+1, n)*chen(k), v=0..k), k=0..n) end:
The diagonals in the full triangle (with zero coefficients) of the polynomials have the general form E(k)*C(n+k, k) (k>=0 fixed, n=0, 1, ...) where E(n) are the Euler number in the enumeration A122045 and the C(n, k) are the binomial coefficients A007318. In Maple parlance: seq(euler(k)*binomial(n+k, k), n=0, ..). For k=2 we find the triangular numbers A000217 and for k=4 the numbers A154286. [From Peter Luschny (peter(AT)luschny.de), Jan 06 2009]
Contribution from Peter Luschny (peter(AT)luschny.de), Jul 07 2009: (Start)
# Coefficients with zeros:
seq(print(seq(coeff(i!*coeff(series(exp(x*t)*sech(t), t, 16), t, i), x, i-n), n=0..i)), i=0..8); (End)
|
