cp's OEIS Frontend

A060083 Coefficients of even-indexed Euler polynomials (rising powers without zeros).

%I A060083 #26 May 08 2018 15:11:55
%S A060083 1,-1,1,1,-2,1,-3,5,-3,1,17,-28,14,-4,1,-155,255,-126,30,-5,1,2073,
%T A060083 -3410,1683,-396,55,-6,1,-38227,62881,-31031,7293,-1001,91,-7,1,
%U A060083 929569,-1529080,754572,-177320,24310,-2184,140,-8,1,-28820619
%N A060083 Coefficients of even-indexed Euler polynomials (rising powers without zeros).
%C A060083 E(2*n,1/2)*(-4)^n = A000364(n) (signless Euler numbers without zeros).
%D A060083 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.
%H A060083 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A060083 J. Cigler, <a href="http://arxiv.org/abs/0908.1219">q-Fibonacci polynomials and q-Genocchi numbers</a>, arXiv:0908.1219
%H A060083 H.-C. Herbig, D. Herden, C. Seaton, <a href="http://arxiv.org/abs/1404.1022">On compositions with x^2/(1-x)</a>, arXiv preprint arXiv:1404.1022, 2014
%F A060083 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.
%F A060083 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
%F A060083 T(n,k) = 1/(2*k+1)*binomial(2*n,2*k)*A001469(n-k) for 0 <= k <= n-1.
%F A060083 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
%t A060083 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 *)
%o A060083 (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)
%Y A060083 A060082 (falling powers).
%Y A060083 Matrix inverse is A102054. Column 0 is A001469 (Genocchi numbers).
%Y A060083 Cf. A102054, A001469.
%K A060083 sign,easy,tabl
%O A060083 0,5
%A A060083 _Wolfdieter Lang_, Mar 29 2001