0,5

Definition. Let the polynomials V_n(x) (n>=0) defined by V_n(x) = sum_{k=0..n} sum_{v=0..k} (-1)^v*C(k,v)*L(k)*(x+v+1)^n; The sequence L(k)=-1-H(k-1)*(-1)^floor((k-1)/4) / 2^floor(k/2) if k>0 and L(0)=0; H(k) = 1 if k mod 4 <> 0 otherwise 0.

(1) V_n(0) = 2^n euler(n,1) for n > 0, A155585.

(2) V_n(1) = 1 - euler(n).

(3) V_{n-1}(0) n / (4^n - 2^n) = B_n gives for n > 1 the Bernoulli numbers A027641/A027642.

(4) V_{n-1}(0) n (2/2^n-2)/(2^n-1) = G_n the Genocchi number A036968 for n > 1.

(5) V_n(1/2)2^{n} - 1 is a signed version of the generalized Euler (Springer) numbers, see A001586.

The Swiss-Knife polynomials (A153641) are complementary to the polynomials defined here. Adding both gives polynomials with e.g.f. exp(x*t)*(sech(t)+tanh(t)), the coefficients of which are a signed variant of A109449.

The Swiss-Knife polynomials as well as the complementary Swiss-Knife polynomials are closely related to the Bernoulli and Euler polynomials. Let F be a sequence and

P_{F}[n](x) = sum_{k=0..n} sum_{v=0..k} (-1)^v*C(k,v)*F(k)*(x+v+1)^n.

V_n(x) = P_{F}[n](x) with F(k)=L(k) defined above, are the Co-Swiss-Knife polynomials,

W_n(x) = P_{F}[n](x) with F(k)=c(k) the Chen sequence defined in A153641 are the Swiss-Knife polynomials.

B_n(x) = P_{F}[n](x-1) with F(k)=1/(k+1) are the Bernoulli polynomials,

E_n(x) = P_{F}[n](x-1) with F(k)=2^(-k) are the Euler polynomials.

The most striking formal difference between the Swiss-Knife-type polynomials and the Bernoulli-Euler type polynomials is: The SK-type polynomials have integer coefficients whereas the BE-type polynomials have rational coefficients.

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.

Peter Luschny, The Swiss-Knife polynomials.

Wikipedia, Bernoulli number.

E.g.f. exp(x*t)tanh(t) = 0*(t^0/0!)+1*(t^1/1!)+(2*x)*(t^2/2!)+(3*x^2-2)*(t^3/3!)+ ...

V_n(x) = -x^n + Sum_{k=0..n} C(n,k)Euler(k)(x+1)^(n-k)

# Polynomials V_n(x):

V := proc(n, x) local k, pow; pow := (n, k) -> `if`(n=0 and k=0, 1, n^k); add(binomial(n, k)*euler(k)*pow(x+1, n-k), k=0..n) - pow(x, n) end:

# Coefficients a(n):

seq(print(seq(coeff(n!*coeff(series(exp(x*t)*tanh(t), t, 16), t, n), x, n-k), k=0..n)), n=0..8);

V_n(k), n=0, 1, .., k=0: A155585, k=1: A009832,

V_n(k), k=0, 1, .., V_0: A000004, V_1: A000012, V_2: A005843, V_3: A100536

Cf. A153641, A154341, A154342, A154343, A154344, A154345.

Sequence in context: A097974 A139036 A159006 this_sequence A090330 A132747 A053399

Adjacent sequences: A162657 A162658 A162659 this_sequence A162661 A162662 A162663

sign,tabf

Peter Luschny (peter(AT)luschny.de), Jul 09 2009