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COMMENTS
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Compare the first and 2nd binomial transforms of this sequence:
first binomial={1,1,-2,1,4,1,-62,1,1384,1,-50522,1,2702764,..};
2nd binomial={1,2,1,-1,1,17,1,-271,1,7937,1,-353791,..};
to that of the first and 2nd binomial transforms of A090145:
first binomial of A090145={1,0,1,-3,1,15,1,-273,1,7935,1,..};
2nd binomial of A090145={1,1,2,1,-4,1,62,1,-1384,1,50522,..}.
Comparison reveals this e.g.f. relation of the two sequences:
e.g.f.: exp(x)*G090158(x) + exp(2x)*G090145(x) = 2 + 2*sinh(x);
e.g.f.: exp(2*x)*G090158(x) - exp(x)*G090145(x) = 2*sinh(x);
thus G090158(x) = 2*(1+sinh(x) + exp(x)*sinh(x))/(exp(x)*(1+exp(2*x)))
G090145(x) = 2*((1+sinh(x))*exp(x) - sinh(x))/(exp(x)*(1+exp(2*x))).
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