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%I #22 Aug 23 2020 06:33:16
%S 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,
%T 0,8,-28,0,70,0,-84,0,17,9,-36,0,126,0,-252,0,153,0,10,-45,0,210,0,
%U -630,0,765,0,-155,11,-55,0,330,0,-1386,0,2805,0,-1705,0,12,-66,0,495
%N Triangle read by rows: coefficients of Genocchi polynomials G(n,x); n times the Euler polynomials.
%C The Genocchi numbers A001489 appear as constant term of every second polynomial and as the negative sum of its coefficients.
%D Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math., 2n-d ed.; Addison-Wesley, 1994, pp. 573-574.
%H D. Dumont and J. Zeng, <a href="http://math.univ-lyon1.fr/homes-www/zeng/public_html/paper/publication.html">Polynomes d'Euler et les fractions continues de Stieltjes-Rogers</a>, Ramanujan J. 2 (1998) 3, 387-410.
%F E.g.f.: Sum_{n >= 1} G(n, x)*t^n/n! = 2*t*e^(x*t)/(1 + e^t).
%F G(n, x) = Sum_{k=1..n} k*C(n, k)* Euler(k-1, 0)*x^(n-k). - _Peter Luschny_, Jul 13 2009
%F 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
%e G(1,x) = 1
%e G(2,x) = 2*x - 1
%e G(3,x) = 3*x^2 - 3*x
%e G(4,x) = 4*x^3 - 6*x^2 + 1
%e G(5,x) = 5*x^4 - 10*x^3 + 5*x
%e G(6,x) = 6*x^5 - 15*x^4 + 15*x^2 - 3
%e G(7,x) = 7*x^6 - 21*x^5 + 35*x^3 - 21*x
%p 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;
%p seq(print(sort(p(n,x))),n=1..8); # _Peter Luschny_, Jul 07 2009
%t 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_ *)
%o (PARI) G(n)=subst(polcoeff(serlaplace(2*x*exp(x*y)/(exp(x)+1)),n),y,x)
%Y A001489(n) = G(2n, 0) = -G(2n, 1). Cf. A081733.
%Y Cf. A060096/A060097, A027641/A027642.
%K tabl,sign,easy
%O 1,2
%A _Ralf Stephan_, Sep 08 2004
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