%I #88 Jun 03 2020 07:42:03

%S 1,1,1,-1,1,-3,1,-6,5,1,-10,25,1,-15,75,-61,1,-21,175,-427,1,-28,350,

%T -1708,1385,1,-36,630,-5124,12465,1,-45,1050,-12810,62325,-50521,1,

%U -55,1650,-28182,228525,-555731,1,-66,2475,-56364,685575,-3334386,2702765,1

%N Nonzero coefficients of the Swiss-Knife polynomials for the computation of Euler, tangent, and Bernoulli numbers (triangle read by rows).

%C In the following the expression [n odd] is 1 if n is odd, 0 otherwise.

%C (+) W_n(0) = E_n are the Euler (or secant) numbers A122045.

%C (+) W_n(1) = T_n are the signed tangent numbers, see A009006.

%C (+) W_{n-1}(1) n / (4^n - 2^n) = B_n gives for n > 1 the Bernoulli number A027641/A027642.

%C (+) W_n(-1) 2^{-n}(n+1) = G_n the Genocchi number A036968.

%C (+) W_n(1/2) 2^{n} are the signed generalized Euler (Springer) number, see A001586.

%C (+) | W_n([n odd]) | the number of alternating permutations A000111.

%C (+) | W_n([n odd]) / n! | for 0<=n the Euler zeta number A099612/A099617 (see Wikipedia on Bernoulli number). - _Peter Luschny_, Dec 29 2008

%C The diagonals in the full triangle (with zero coefficients) of the polynomials have the general form E(k)*binomial(n+k,k) (k>=0 fixed, n=0,1,...) where E(n) are the Euler numbers in the enumeration A122045. For k=2 we find the triangular numbers A000217 and for k=4 A154286. - _Peter Luschny_, Jan 06 2009

%C From _Peter Bala_, Jun 10 2009: (Start)

%C The Swiss-Knife polynomials W_n(x) may be expressed in terms of the Bernoulli polynomials B(n,x) as

%C ... W_n(x) = 4^(n+1)/(2*n+2)*[B(n+1,(x+3)/4) - B(n+1,(x+1)/4)].

%C The Swiss-Knife polynomials are, apart from a multiplying factor, examples of generalized Bernoulli polynomials.

%C 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 generalized Bernoulli polynomials B(X;n,x), n = 1,2,..., associated with the character X are defined by means of the generating function

%C ... 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 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 In general, W_n(x) = -2/(n+1)*B(X;n+1,x).

%C For the theory of generalized Bernoulli polynomials associated to a periodic arithmetical function see [Cohen, Section 9.4].

%C The generalized Bernoulli polynomials may be used to evaluate twisted sums of k-th powers. For the present case the result is

%C sum{n = 0..4*N-1} X(n)*n^k = 1^k - 3^k + 5^k - 7^k + ... - (4*N-1)^k

%C = [B(X;k+1,4*N) - B(X;k+1,0)]/(k+1) = [W_k(0) - W_k(4*N)]/2.

%C For the proof apply [Cohen, Corollary 9.4.17 with m = 4 and x = 0].

%C The generalized 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 generalized Bernoulli polynomials attached to a Dirichlet character modulus 8 see A151751.

%C (End)

%C 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

%C 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:

%C gcd(coeffs(SKP_{n}(x) - x^n)) = A155457(n) (n>1). - _Peter Luschny_, Dec 16 2009

%C Another version is at A119879. - _Philippe Deléham_, Oct 26 2013

%D H. Cohen, Number Theory - Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics. Springer-Verlag. [From Peter Bala, Jun 10 2009]

%H G. C. Greubel, <a href="/A153641/b153641.txt">Table of n, a(n) for the first 76 rows, flattened</a>

%H Kwang-Wu Chen, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/CHEN/AlgBE2.html">Algorithms for Bernoulli numbers and Euler numbers</a>, J. Integer Sequences, 4 (2001), #01.1.6.

%H Suyoung Choi and Hanchul Park, <a href="http://arxiv.org/abs/1210.3776">A new graph invariant arises in toric topology</a>, arXiv preprint arXiv:1210.3776 [math.AT], 2012.

%H Leonhard Euler (1735), <a href="http://arxiv.org/abs/math/0506415">De summis serierum reciprocarum</a>, Opera Omnia I.14, E 41, 73-86; On the sums of series of reciprocals, arXiv:math/0506415v2 (math.HO), 2005-2008.

%H A. Hodges and C. V. Sukumar, <a href="http://dx.doi.org/10.1098/rspa.2007.0001">Bernoulli, Euler, permutations and quantum algebras</a>, Proc. R. Soc. A Oct. 2007 vol 463 no. 463 2086 2401-2414. [Added by Tom Copeland, Aug 31 2015]

%H Peter Luschny, <a href="http://www.luschny.de/math/seq/SwissKnifePolynomials.html">The Swiss-Knife polynomials.</a>

%H Peter Luschny, <a href="http://www.oeis.org/wiki/User:Peter_Luschny/SwissKnifePolynomials">Swiss-Knife polynomials and Euler numbers</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Bernoulli_number">Bernoulli number</a>

%H J. Worpitzky, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002158698">Studien über die Bernoullischen und Eulerschen Zahlen</a>, Journal für die reine und angewandte Mathematik, 94 (1883), 203-232.

%F W_n(x) = Sum_{k=0..n}{v=0..k} (-1)^v binomial(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).

%F From _Peter Bala_, Jun 10 2009: (Start)

%F 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! + ....

%F W_n(x) = 1/(2*n+2)*Sum_{k=0..n+1} 1/(k+1)*Sum_{i=0..k} (-1)^i*binomial(k,i)*((x+4*i+3)^(n+1) - (x+4*i+1)^(n+1)).

%F Fourier series expansion for the generalized Bernoulli polynomials:

%F 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 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 (End)

%F E.g.f.: exp(x*t) * sech(t). - _Peter Luschny_, Jul 07 2009

%F O.g.f. as a J-fraction: z/(1-x*z+z^2/(1-x*z+4*z^2/(1-x*z+9*z^2/(1-x*z+...)))) = z + x*z^2 + (x^2-1)*z^3 + (x^3-3*x)*z^4 + .... - _Peter Bala_, Mar 11 2012

%F Conjectural o.g.f.: Sum_{n >= 0} 1/2^(n-1)/2)*cos((n+1)*Pi/4)*( Sum_{k = 0..n} (-1)^k*binomial(n,k)/(1 - (k + x)*t) ) = 1 + x*t + (x^2 - 1)*t^2 + (x^3 - 3*x)*t^3 + ... (checked up to O(t^13)), which leads to W_n(x) = Sum_{k = 0..n} 1/2^((k - 1)/2)*cos((k + 1)*Pi/4)*( Sum_{j = 0..k} (-1)^j*binomial(k, j)*(j + x)^n ). - _Peter Bala_, Oct 03 2016

%e 1

%e x

%e x^2 -1

%e x^3 -3x

%e x^4 -6x^2 +5

%e x^5 -10x^3 +25x

%e x^6 -15x^4 +75x^2 -61

%e x^7 -21x^5 +175x^3 -427x

%p 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 # Coefficients with zeros:

%p 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);

%p # Recursion

%p W := proc(n,z) option remember; local k,p;

%p if n = 0 then 1 else p := irem(n+1,2);

%p z^n - p + add(`if`(irem(k,2)=1,0,

%p W(k,0)*binomial(n,k)*(power(z,n-k)-p)),k=2..n-1) fi end:

%p # _Peter Luschny_, edited and additions Jul 07 2009, May 13 2010, Oct 24 2011

%t 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] (* _Jean-François Alcover_, Oct 03 2011, after g.f. *)

%t sk[n_, x_] := Sum[Binomial[n, k]*EulerE[k]*x^(n-k), {k, 0, n}]; Table[CoefficientList[sk[n, x], x] // Reverse // Select[#, # =!= 0 &] &, {n, 0, 13}] // Flatten (* _Jean-François Alcover_, May 21 2013 *)

%o (Sage)

%o def A046978(k):

%o if k % 4 == 0:

%o return 0

%o return (-1)**(k // 4)

%o def A153641_poly(n, x):

%o 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)))

%o for n in (0..7): print(A153641_poly(n, x)) # _Peter Luschny_, Oct 24 2011

%Y Cf. A151751, A162590, A000111, A001586, A009006, A027641/A027642, A036968, A099612/A099617, A119879, A104033.

%Y W_n(k), k=0,1,...

%Y W_0: 1, 1, 1, 1, 1, 1, ........ A000012

%Y W_1: 0, 1, 2, 3, 4, 5, ........ A001477

%Y W_2: -1, 0, 3, 8, 15, 24, ........ A067998

%Y W_3: 0, -2, 2, 18, 52, 110, ........ A121670

%Y W_4: 5, 0, -3, 32, 165, 480, ........

%Y W_n(k), n=0,1,...

%Y k=0: 1, 0, -1, 0, 5, 0, -61, ... A122045

%Y k=1: 1, 1, 0, -2, 0, 16, 0, ... A155585

%Y k=2: 1, 2, 3, 2, -3, 2, 63, ... A119880

%Y k=3: 1, 3, 8, 18, 32, 48, 128, ... A119881

%Y k=4: 1, 4, 15, 52, 165, 484, ........ [_Peter Luschny_, Jul 07 2009]

%K easy,sign,tabf

%O 0,6

%A _Peter Luschny_, Dec 29 2008