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EXAMPLE
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Successive binomial transforms are:
0th: {1,0,-8,24,64,-480,-3968,34944,354304,-4062720,...}
1st: {1,1,-7,1,113,1,-5527,1,501473,1,-73163047,1,...}
2nd: {1,2,-4,-16,80,512,-3904,-34816,354560,4063232,...}
3rd: {1,3,1,-21,1,723,1,-49221,1,5746083, 1,...} and
4th: {1,4,8,-8,-64,544,3968, -34688,-354304,4063744,...}
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).
This describes the e.g.f.: A+exp(4x)*A=2+2sinh(2x).
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