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A153641
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Nonzero coefficients of the Swiss-Knife polynomials for the computation of Euler, tangent and Bernoulli number (triangle read by rows).
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20
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1, 1, 1, -1, 1, -3, 1, -6, 5, 1, -10, 25, 1, -15, 75, -61, 1, -21, 175, -427, 1, -28, 350, -1708, 1385, 1, -36, 630, -5124, 12465, 1, -45, 1050, -12810, 62325, -50521, 1, -55, 1650, -28182, 228525, -555731, 1, -66, 2475, -56364, 685575, -3334386, 2702765, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| 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). [Peter Luschny, Dec 29 2008]
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 numbers in the enumeration A122045 and the C(n,k) are the binomial coefficients A007318. For k=2 we find the triangular numbers A000217 and for k=4 A154286. [Peter Luschny, Jan 06 2009]
Contribution from Peter Bala, 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)
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. [Peter Luschny, Jul 12 2009]
The greatest common divisor of the nonzero coefficients of the decapitated Swiss-Knife polynomials is exp(Lambda(n)), where Lambda(n) is the von Mangoldt function for odd primes, symbolically:
gcd(coeffs(SKP_{n}(x) - x^n)) = A155457(n) (n>1). [Peter Luschny, Dec 16 2009]
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REFERENCES
| Kwang-Wu Chen, Algorithms for Bernoulli numbers and Euler numbers, Journal of Integer Sequences, 4 (2001), [01.1.6].
H. Cohen, Number Theory - Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics. Springer-Verlag. [From Peter Bala, Jun 10 2009]
Euler, Leonhard (1735), "De summis serierum reciprocarum", Opera Omnia I.14, E 41, 73-86; On the sums of series of reciprocals, arXiv:math/0506415v2 (math.HO).
J. Worpitzky, Studien ueber die Bernoullischen und Eulerschen Zahlen, Journal fuer die reine und angewandte Mathematik, 94 (1883), 203--232.
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LINKS
| Peter Luschny, The Swiss-Knife polynomials.
Peter Luschny, Swiss-Knife polynomials and Euler numbers
Wikipedia, Bernoulli number.
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FORMULA
| w_n(x) = sum_{k=0}^{n} sum_{v=0}^{k} (-1)^{v} binom{k}{v} c_k (x+v+1)^n where
c_k = frac{(-1)^{floor(k/4)}{2^{floor(k/2)}} [4 not div k] (Iverson notation)
Contribution from Peter Bala, Jun 10 2009: (Start)
E.g.f.:
2*exp(x*t)*(exp(t)-exp(3*t))/(1-exp(4*t))= 1 + x*t + (x^2-1)*t^2/2! + (x^3-3*x)*t^3/3! + ....
W_n(x) = 1/(2*n+2)*sum {k = 0..n+1} 1/(k+1)*sum {i = 0..k} (-1)^i*comb(k,i)*{(x+4*i+3)^(n+1) - (x+4*i+1)^(n+1)}.
Fourier series expansion for the generalised Bernoulli polynomials:
B(X;2*n,x) = (-1)^n*(2/Pi)^(2*n)*(2*n)! * {sin(Pi*x/2)/1^(2*n) - sin(3*Pi*x/2)/3^(2*n) + sin(5*Pi*x/2)/5^(2*n) - ...}, valid for 0 <= x <= 1 when n >= 1.
B(X;2*n+1,x) = (-1)^(n+1)*(2/Pi)^(2*n+1)*(2*n+1)! * {cos(Pi*x/2)/1^(2*n+1) - cos(3*Pi*x/2)/3^(2*n+1) + cos(5*Pi*x/2)/5^(2*n+1) - ...}, valid for 0 <= x <= 1 when n >= 1 and for 0 <= x < 1 when n = 0.
(End)
E.g.f.: exp(x*t)sech(t). [Peter Luschny, Jul 07 2009]
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EXAMPLE
| 1
x
x^2 -1
x^3 -3x
x^4 -6x^2 +5
x^5 -10x^3 +25x
x^6 -15x^4 +75x^2 -61
x^7 -21x^5 +175x^3 -427x
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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:
# 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);
# Recursion
W := proc(n, z) option remember; local k, p;
if n = 0 then 1 else p := irem(n+1, 2);
z^n - p + add(`if`(irem(k, 2)=1, 0,
W(k, 0)*binomial(n, k)*(power(z, n-k)-p)), k=2..n-1) fi end:
- Peter Luschny, edited and additions Jul 07 2009, May 13 2010, Oct 24 2011.
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MATHEMATICA
| max = 9; rows = (Reverse[ CoefficientList[ #, x]] & ) /@ CoefficientList[ Series[ Exp[x*t]*Sech[t], {t, 0, max}], t]*Range[0, max]!; par[coefs_] := (p = Partition[ coefs, 2][[All, 1]]; If[ EvenQ[ Length[ coefs]], p, Append[ p, Last[ coefs]]]); Flatten[ par /@ rows] (* From Jean-François Alcover, Oct 03 2011, after g.f. *)
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PROG
| (SAGE)
def A046978(k) :
if k % 4 == 0 :
return 0
else:
return (-1)^(k//4)
def A153641_poly(n, x) :
return expand(add(2^(-(k//2))*A046978(k+1)*add((-1)^v*binomial(k, v)*(v+x+1)^n for v in (0..k)) for k in (0..n)))
for n in (0..7) : print A153641_poly(n, x) ## Peter Luschny, Oct 24 2011
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CROSSREFS
| Cf. A151751 [From Peter Bala, Jun 10 2009], A162590.
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, ..... [Peter Luschny, Jul 07 2009]
Sequence in context: A112351 A143858 A109954 * A133545 A153091 A179069
Adjacent sequences: A153638 A153639 A153640 * A153642 A153643 A153644
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KEYWORD
| easy,sign,tabf
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AUTHOR
| Peter Luschny (peter(AT)luschny.de), Dec 29 2008
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EXTENSIONS
| Cross-references corrected by Peter Bala (pbala(AT)talktalk.net), Jun 19 2009
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