answer to Wouter's question

[N. J. A. Sloane]

Superseeker is telling you that if G is
the g.f. for your sequence, then 1/G has coefficients
that match those two sequences. 

And that is true, look:

> s1:=[1,1,3,6,15,31,73,157,358,784];

G is

1+1*x+3*x^2+6*x^3+15*x^4+31*x^5+73*x^6+157*x^7+358*x^8+784*x^9+O(x^10)

1/G is

1-1*x-2*x^2-1*x^3-2*x^4+1*x^5-1*x^6+3*x^7+5*x^9+O(x^10)

Neil