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 A098434 Triangle read by rows: coefficients of Genocchi polynomials G(n,x); n times the Euler polynomials. 0
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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

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Last modified March 21 14:15 EDT 2023. Contains 361408 sequences. (Running on oeis4.)