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A098434
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Triangle read by rows: coefficients of Genocchi polynomials G(n,x); n times the Euler polynomials.
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0
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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
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OFFSET
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1,2
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COMMENTS
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The Genocchi numbers A001489 appear as constant term of every second polynomial and as the negative sum of its coefficients.
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REFERENCES
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Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math., 2n-d ed.; Addison-Wesley, 1994, pp. 573-574.
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LINKS
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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.
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FORMULA
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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
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EXAMPLE
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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
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(PARI) G(n)=subst(polcoeff(serlaplace(2*x*exp(x*y)/(exp(x)+1)), n), y, x)
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CROSSREFS
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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
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KEYWORD
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tabl,sign,easy
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AUTHOR
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Ralf Stephan, Sep 08 2004
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STATUS
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approved
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