A098434 Triangle read by rows: coefficients of Genocchi polynomials G(n,x); n times the Euler polynomials.

1, 2, -1, 3, -3, 0, 4, -6, 0, 1, 5, -10, 0, 5, 0, 6, -15, 0, 15, 0, -3, 7, -21, 0, 35, 0, -21, 0, 8, -28, 0, 70, 0, -84, 0, 17, 9, -36, 0, 126, 0, -252, 0, 153, 0, 10, -45, 0, 210, 0, -630, 0, 765, 0, -155, 11, -55, 0, 330, 0, -1386, 0, 2805, 0, -1705, 0, 12, -66, 0, 495

Offset: 1

Author

Ralf Stephan, Sep 08 2004

Keywords

• tabl
• sign
• easy

Comments

The Genocchi numbers A001489 appear as constant term of every second polynomial and as the negative sum of its coefficients.

Examples

			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
		

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.

Crossrefs

A001489(n) = G(2n, 0) = -G(2n, 1). Cf. A081733.

Cf. A060096/A060097, A027641/A027642.

Programs

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 *)

PARI

G(n)=subst(polcoeff(serlaplace(2*x*exp(x*y)/(exp(x)+1)),n),y,x)

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