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 A060083 Coefficients of even-indexed Euler polynomials (rising powers without zeros). 7
 1, -1, 1, 1, -2, 1, -3, 5, -3, 1, 17, -28, 14, -4, 1, -155, 255, -126, 30, -5, 1, 2073, -3410, 1683, -396, 55, -6, 1, -38227, 62881, -31031, 7293, -1001, 91, -7, 1, 929569, -1529080, 754572, -177320, 24310, -2184, 140, -8, 1, -28820619 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS E(2*n,1/2)*(-4)^n = A000364(n) (signless Euler numbers without zeros). REFERENCES M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 809. LINKS M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. J. Cigler, q-Fibonacci polynomials and q-Genocchi numbers, arXiv:0908.1219 H.-C. Herbig, D. Herden, C. Seaton, On compositions with x^2/(1-x), arXiv preprint arXiv:1404.1022, 2014 FORMULA E(2*n, x)= sum(a(n, m)*x^(2*m+1), m=0..n-1) + x^(2*n), n >= 1; E(0, x)=1. T(n, k) = A102054(n, k+1) - A102054(n+1, k+1), where A102054 is matrix inverse. E.g.f.: A(x^2, y^2) = [cosh(xy)*(y-1) + exp(xy)/(exp(x)+1) + exp(-xy)/(exp(-x)+1)]/y. - Paul D. Hanna, Dec 28 2004 T(n,k) = 1/(2*k+1)*binomial(2*n,2*k)*A001469(n-k) for 0 <= k <= n-1. Let F(n,x) = Sum_{k=0..n-1} binomial(n-k-1,k)*x^k be a Fibonacci polynomial (see A011973 for coefficients). Then F(2*n,x) = -Sum_{k=0..n-1} T(n,k)*F(2*k+1,x). For example, F(8,x) = -17*F(1,x) + 28*F(3,x) - 14*F(5,x) + 4*F(7,x). See Cigler, Corollary 1.3. - Peter Bala, Mar 14 2012 MATHEMATICA t[n_, k_] := Binomial[2*n, 2*k]*2*(n - k)*EulerE[2*(n - k) - 1, 0]/(2*k + 1); t[n_, n_] = 1; Table[t[n, k], {n, 0, 9}, {k, 0, n }] // Flatten (* Jean-François Alcover, Jul 03 2013 *) PROG (PARI) {T(n, k)=local(X=x+x*O(x^(2*n)), Y=y+y*O(y^(2*k+1))); (2*n)!*polcoeff(polcoeff((cosh(X*Y)*(Y-1)+ exp(X*Y)/(exp(X)+1)+exp(-X*Y)/(exp(-X)+1))/Y, 2*n, x), 2*k, y)} (Hanna) CROSSREFS A060082 (falling powers). Matrix inverse is A102054. Column 0 is A001469 (Genocchi numbers). Cf. A102054, A001469. Sequence in context: A193953 A201377 A322942 * A069931 A209152 A209158 Adjacent sequences:  A060080 A060081 A060082 * A060084 A060085 A060086 KEYWORD sign,easy,tabl AUTHOR Wolfdieter Lang, Mar 29 2001 STATUS approved

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Last modified June 19 23:39 EDT 2019. Contains 324222 sequences. (Running on oeis4.)