A001037, A059966, A060477

[Michael Somos]

Eric Weisstein asked :

> Is it necessary to keep these three as separate sequences?  Modulo the 1st 

The answer is yes. The reason has to do with the role of these
sequences with other sequences among other things. For example :

? print(EULER0([2, 1, 2, 3, 6, 9, 18, 30, 56, 99]))
[1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024]
? print(EULER0([1, 1, 2, 3, 6, 9, 18, 30, 56, 99]))
[1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512]

So that, as noted in OEIS, EULER(A001037)=A000079, and also
EULER(A059966)=A011782 (not noted). The difference between
A001037(1)=2 and A059966(1)=1 results in an extra 1/(1-x)
factor which is the relation between A000079 and A011782.
These last two sequences are essentially the same also, but
there is an offset shift here and an extra A011782(0)=1.
Shalom, Michael