OFFSET
1,2
COMMENTS
The Genocchi numbers A001489 appear as constant term of every second polynomial and as the negative sum of its coefficients.
REFERENCES
Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math., 2n-d ed.; Addison-Wesley, 1994, pp. 573-574.
LINKS
D. Dumont and J. Zeng, Polynomes d'Euler et les fractions continues de Stieltjes-Rogers, Ramanujan J. 2 (1998) 3, 387-410.
FORMULA
E.g.f.: Sum_{n >= 1} G(n, x)*t^n/n! = 2*t*e^(x*t)/(1 + e^t).
G(n, x) = Sum_{k=1..n} k*C(n, k)* Euler(k-1, 0)*x^(n-k). - Peter Luschny, Jul 13 2009
G(n, x) = n*Euler(n-1,x) = Sum_{k=0..n} binomial(n,k)*Bernoulli(k)*2*(1-2^k)*x^(n-k), with the Euler polynomials Euler(n,x) (see A060096/A060097) and Bernoulli numbers A027641/A027642. See the Graham et al. reference, pp. 573-574, Exercise 7.52. - Wolfdieter Lang, Mar 13 2017
EXAMPLE
G(1,x) = 1
G(2,x) = 2*x - 1
G(3,x) = 3*x^2 - 3*x
G(4,x) = 4*x^3 - 6*x^2 + 1
G(5,x) = 5*x^4 - 10*x^3 + 5*x
G(6,x) = 6*x^5 - 15*x^4 + 15*x^2 - 3
G(7,x) = 7*x^6 - 21*x^5 + 35*x^3 - 21*x
MAPLE
p := proc(n, x) local j, k; add(binomial(n, k)*add(binomial(k, j)*2^j*bernoulli(j), j=0..k-1)*x^(n-k), k=0..n) end;
seq(print(sort(p(n, x))), n=1..8); # Peter Luschny, Jul 07 2009
MATHEMATICA
g[n_, x_] := Sum[ k Binomial[n, k] EulerE[k-1, 0] x^(n-k), {k, 1, n}]; Table[ CoefficientList[g[n, x], x] // Reverse, {n, 1, 12}] // Flatten (* Jean-François Alcover, May 23 2013, after Peter Luschny *)
PROG
(PARI) G(n)=subst(polcoeff(serlaplace(2*x*exp(x*y)/(exp(x)+1)), n), y, x)
CROSSREFS
KEYWORD
AUTHOR
Ralf Stephan, Sep 08 2004
STATUS
approved