%I A153641
%S A153641 1,1,1,1,1,3,1,6,5,1,10,25,1,15,75,61,1,21,175,427,1,28,350,
%T A153641 1708,1385,1,36,630,5124,12465
%V A153641 1,1,1,-1,1,-3,1,-6,5,1,-10,25,1,-15,75,-61,1,-21,175,-427,1,-28,350,
%W A153641 -1708,1385,1,-36,630,-5124,12465
%N A153641 Nonzero coefficients of the Swiss-Knife polynomials for the computation 
               of Euler, tangent and Bernoulli number (triangle read by rows).
%C A153641 In the following the expression [n odd] is 1 if n is odd, 0 otherwise.
%C A153641 (+) W_n(0) = E_n are the Euler (or secant) numbers A122045.
%C A153641 (+) W_n(1) = T_n are the signed tangent numbers, see A009006.
%C A153641 (+) W_{n-1}(1) n / (4^n - 2^n) = B_n gives for n > 1 the Bernoulli number 
               A027641/A027642.
%C A153641 (+) W_n(-1) 2^{-n}(n+1) = G_n the Genocchi number A036968.
%C A153641 (+) W_n(1/2) 2^{n} are the signed generalized Euler (Springer) number, 
               see A001586.
%C A153641 (+) | W_n([n odd]) | the number of alternating permutations A000111.
%C A153641 (+) | W_n([n odd]) / n! | for 0<=n the Euler zeta number A099612/A099617 
               (see Wikipedia on Bernoulli number).
%C A153641 Contribution from Peter Bala (pbala(AT)talktalk.net), Jun 10 2009: (Start)
%C A153641 The Swiss-Knife polynomials W_n(x) may be expressed in terms of the Bernoulli 
               polynomials B(n,x) as
%C A153641 ... W_n(x) = 4^(n+1)/(2*n+2)*[B(n+1,(x+3)/4) - B(n+1,(x+1)/4)].
%C A153641 The Swiss_Knife polynomials are, apart from a multiplying factor, examples 
               of generalised Bernoulli polynomials.
%C A153641 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
%C A153641 ... t*exp(x*t)*(exp(t)-exp(3*t))/(exp(4*t)-1) = sum {n = 1..inf} B(X;
               n,x)*t^n/n!.
%C A153641 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).
%C A153641 In general, W_n(x) = -2/(n+1)*B(X;n+1,x).
%C A153641 For the theory of generalised Bernoulli polynomials associated to a periodic 
               arithmetical function see [Cohen, Section 9.4].
%C A153641 The generalised Bernoulli polynomials may be used to evaluate twisted 
               sums of k-th powers. For the present case the result is
%C A153641 sum{n = 0..4*N-1} X(n)*n^k = 1^k - 3^k + 5^k - 7^k + ... - (4*N-1)^k
%C A153641 = [B(X;k+1,4*N) - B(X;k+1,0)]/(k+1) = [W_k(0) - W_k(4*N)]/2.
%C A153641 For the proof apply [Cohen, Corollary 9.4.17 with m = 4 and x = 0].
%C A153641 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.
%C A153641 (End)
%C A153641 Contribution from Peter Luschny (peter(AT)luschny.de), Jul 12 2009: (Start)
%C A153641 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)
%C A153641 Contribution from Peter Luschny (peter(AT)luschny.de), Jul 07 2009: (Start)
%C A153641 W_n(k), k=0,1,...
%C A153641 W_0: 1, 1, 1, 1, 1, 1, .......... A000012
%C A153641 W_1: 0, 1, 2, 3, 4, 5, .......... A001477
%C A153641 W_2: -1, 0, 3, 8, 15, 24, ....... A067998
%C A153641 W_3: 0, -2, 2, 18, 52, 110, ..... A136434
%C A153641 W_4: 5, 0, -3, 32, 165, 480, ....
%C A153641 W_n(k), n=0,1,...
%C A153641 k=0: 1, 0, -1, 0, 5, 0, -61, ..... A122045
%C A153641 k=1: 1, 1, 0, -2, 0, 16, 0, ...... A155585
%C A153641 k=2: 1, 2, 3, 2, -3, 2, 63, ...... A119880
%C A153641 k=3: 1, 3, 8, 18, 32, 48, 128, ... A119881
%C A153641 k=4: 1, 4, 15, 52, 165, 484, .....
%C A153641 (End)
%D A153641 Kwang-Wu Chen, Algorithms for Bernoulli numbers and Euler numbers, Journal 
               of Integer Sequences, 4 (2001), [01.1.6].
%D A153641 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).
%D A153641 J. Worpitzky, Studien ueber die Bernoullischen und Eulerschen Zahlen, 
               Journal fuer die reine und angewandte Mathematik, 94 (1883), 203--232.
%D A153641 H. Cohen, Number Theory - Volume II: Analytic and Modern Tools, Graduate 
               Texts in Mathematics. Springer-Verlag. [From Peter Bala (pbala(AT)talktalk.net), 
               Jun 10 2009]
%H A153641 Peter Luschny, <a href="http://www.luschny.de/math/seq/SwissKnifePolynomials.html">
               The Swiss-Knife polynomials.</a>
%H A153641 Wikipedia, <a href="http://en.wikipedia.org/wiki/Bernoulli_number">Bernoulli 
               number.</a>
%F A153641 w_n(x) = sum_{k=0}^{n} sum_{v=0}^{k} (-1)^{v} binom{k}{v} c_k (x+v+1)^n 
               where
%F A153641 c_k = frac{(-1)^{floor(k/4)}{2^{floor(k/2)}} [4 not div k] (Iverson notation)
%F A153641 Contribution from Peter Bala (pbala(AT)talktalk.net), Jun 10 2009: (Start)
%F A153641 E.g.f.:
%F A153641 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! + ....
%F A153641 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)}.
%F A153641 Fourier series expansion for the generalised Bernoulli polynomials:
%F A153641 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.
%F A153641 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.
%F A153641 (End)
%F A153641 E.g.f. exp(x*t)sech(t). [From Peter Luschny (peter(AT)luschny.de), Jul 
               07 2009]
%e A153641 1
%e A153641 x
%e A153641 x^2 -1
%e A153641 x^3 -3x
%e A153641 x^4 -6x^2 +5
%e A153641 x^5 -10x^3 +25x
%e A153641 x^6 -15x^4 +75x^2 -61
%e A153641 x^7 -21x^5 +175x^3 -427x
%p A153641 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:
%p A153641 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]
%p A153641 Contribution from Peter Luschny (peter(AT)luschny.de), Jul 07 2009: (Start)
%p A153641 # Coefficients with zeros:
%p A153641 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)
%Y A153641 Cf. A151751. [From Peter Bala (pbala(AT)talktalk.net), Jun 10 2009]
%Y A153641 Cf. A162590 [From Peter Luschny (peter(AT)luschny.de), Jul 07 2009]
%Y A153641 Sequence in context: A112351 A143858 A109954 this_sequence A133545 A153091 
               A124847
%Y A153641 Adjacent sequences: A153638 A153639 A153640 this_sequence A153642 A153643 
               A153644
%K A153641 easy,sign,tabf
%O A153641 0,6
%A A153641 Peter Luschny (peter(AT)luschny.de), Dec 29 2008
%E A153641 Cross-references corrected by Peter Bala (pbala(AT)talktalk.net), Jun 
               19 2009