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

1,2

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

Table of n, a(n) for n=1..70.

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

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

Cf. A060096/A060097, A027641/A027642.

Sequence in context: A184344 A144243 A125210 * A212634 A162883 A081446

Adjacent sequences: A098431 A098432 A098433 * A098435 A098436 A098437

Keyword

tabl,sign,easy

Author

Ralf Stephan, Sep 08 2004

Status

approved