OFFSET
0,5
COMMENTS
Definition. V_n(x) = (skp(n, x+1) - skp(n, x-1))/2 where skp(n,x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Jul 23 2012
Equivalently, 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).
(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.
Let R be the exponential Riordan array (exp(x)*sech(x), x) = P * A119879 = 2*P(I + P^2)^(-1) where P denotes Pascal's triangle A007318. Then T = R - I. - Peter Bala, Mar 07 2024
LINKS
Leonhard Euler (1735), De summis serierum reciprocarum, Opera Omnia I.14, E 41, 73-86; On the sums of series of reciprocals, arXiv:math/0506415 [math.HO], 2005-2008.
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
T(n, k) = [x^(n-k)](skp(n,x+1)-skp(n,x-1))/2) where skp(n,x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Jul 23 2012
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).
EXAMPLE
Triangle begins:
[0] 0;
[1] 1, 0;
[2] 0, 2, 0;
[3] -2, 0, 3, 0;
[4] 0, -8, 0, 4, 0;
[5] 16, 0, -20, 0, 5, 0;
[6] 0, 96, 0, -40, 0, 6, 0;
[7] -272, 0, 336, 0, -70, 0, 7, 0;
[8] 0, -2176, 0, 896, 0, -112, 0, 8, 0;
[9] 7936, 0, -9792, 0, 2016, 0, -168, 0, 9, 0;
MAPLE
# 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, k), k=0..n)), n=0..8);
MATHEMATICA
skp[n_, x_] := Sum[Binomial[n, k]*EulerE[k]*x^(n-k), {k, 0, n}]; v[n_, x_] := (skp[n, x+1]-skp[n, x-1])/2; t[n_, k_] := Coefficient[v[n, x], x, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 09 2014 *)
PROG
(Sage)
R = PolynomialRing(QQ, 'x')
@CachedFunction
def skp(n, x) : # Swiss-Knife polynomials A153641.
if n == 0 : return 1
return add(skp(k, 0)*binomial(n, k)*(x^(n-k)-(n+1)%2) for k in range(n)[::2])
def A162660(n, k) : return 0 if k > n else R((skp(n, x+1)-skp(n, x-1))/2)[k]
matrix(ZZ, 9, A162660) # Peter Luschny, Jul 23 2012
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Jul 09 2009
STATUS
approved