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A153641 Nonzero coefficients of the Swiss-Knife polynomials for the computation of Euler, tangent, and Bernoulli numbers (triangle read by rows). 50
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; text; internal format)
OFFSET

0,6

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)*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

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 generalized 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 generalized 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 generalized Bernoulli polynomials associated to a periodic arithmetical function see [Cohen, Section 9.4].

The generalized 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 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.

(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

Another version is at A119879. - Philippe Deléham, Oct 26 2013

REFERENCES

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

LINKS

G. C. Greubel, Table of n, a(n) for the first 76 rows, flattened

Kwang-Wu Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.

Suyoung Choi and Hanchul Park, A new graph invariant arises in toric topology, arXiv preprint arXiv:1210.3776 [math.AT], 2012.

Leonhard Euler (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), 2005-2008.

A. Hodges and C. V. Sukumar, Bernoulli, Euler, permutations and quantum algebras, Proc. R. Soc. A Oct. 2007 vol 463 no. 463 2086 2401-2414 [Added by Tom Copeland, Aug 31 2015]

Peter Luschny, The Swiss-Knife polynomials.

Peter Luschny, Swiss-Knife polynomials and Euler numbers

Wikipedia, Bernoulli number

J. Worpitzky, Studien über die Bernoullischen und Eulerschen Zahlen, Journal für die reine und angewandte Mathematik, 94 (1883), 203-232.

FORMULA

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).

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*binomial(k,i)*((x+4*i+3)^(n+1) - (x+4*i+1)^(n+1)).

Fourier series expansion for the generalized 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

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

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

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

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

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] (* Jean-François Alcover, Oct 03 2011, after g.f. *)

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 *)

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

CROSSREFS

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

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, ........ A121670

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: A143858 A258993 A109954 * A133545 A210214 A322427

Adjacent sequences:  A153638 A153639 A153640 * A153642 A153643 A153644

KEYWORD

easy,sign,tabf

AUTHOR

Peter Luschny, Dec 29 2008

STATUS

approved

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Last modified December 10 14:27 EST 2019. Contains 329896 sequences. (Running on oeis4.)