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%I #16 Jan 04 2021 20:01:08
%S 1,0,-8,24,64,-480,-3968,34944,354304,-4062720,-51734528,724568064,
%T 11070521344,-183240744960,-3266330329088,62382319632384,
%U 1270842139869184,-27507470234419200,-630424777639067648,15250953398036987904,388362339077349965824
%N Odd-indexed terms of the first binomial transform equals 1 and the even-indexed terms of the third binomial transform equals 1, with a(0)=1.
%F E.g.f.: 2*(1+sinh(2*x))/(1+exp(4*x)).
%F a(n) ~ n! * (cos(Pi*n/2)-sin(Pi*n/2)) / (Pi/4)^(n+1). - _Vaclav Kotesovec_, Mar 06 2014
%F a(n) = 2^(n-1)*(EulerE(n) - 2^n*(EulerE(n,-1/2) - 2*EulerE(n,0))). - _Benedict W. J. Irwin_, May 26 2016
%e Successive binomial transforms are:
%e 0th: {1,0,-8,24,64,-480,-3968,34944,354304,-4062720,...}
%e 1st: {1,1,-7,1,113,1,-5527,1,501473,1,-73163047,1,...}
%e 2nd: {1,2,-4,-16,80,512,-3904,-34816,354560,4063232,...}
%e 3rd: {1,3,1,-21,1,723,1,-49221,1,5746083, 1,...} and
%e 4th: {1,4,8,-8,-64,544,3968, -34688,-354304,4063744,...}
%e The sum of this sequence with its 4th binomial transform equals {2,4,0,16,0,64,0,64,0,256,0,1024,...}, which has e.g.f.: 2+2sinh(2x).
%e This describes the e.g.f.: A+exp(4x)*A=2+2sinh(2x).
%t CoefficientList[Series[2*(1+Sinh[2*x])/(1+E^(4*x)), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Mar 06 2014 *)
%t Table[2^(n - 1)*(EulerE[n]-2^n (EulerE[n, -1/2] - 2 EulerE[n, 0])), {n, 0, 20}] (* _Benedict W. J. Irwin_, May 26 2016 *)
%Y Cf. A090145, A090158.
%K sign
%O 0,3
%A _Paul D. Hanna_, Nov 25 2003
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