For a sequence of numbers, the generating function has the form $$ g(x)=\sum_{n\space\geq\space0}a_nx^n $$ If $n$ starts from $2$ instead, should $\displaystyle g(x)=\sum_{n\space\geq\space2}a_{n+2}x^{n+2} $, (or $\displaystyle g(x)=\sum_{n\space\geq\space0}a_{n+1}x^{n+1}-1-a_1x$)? I'm not super clear if where n starts is conventionally chosen as $0$.
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- If you used your second form, you'd have $g(x)=a_4 x^4+a^5x^5+\cdots $ which is presumably not what you want. If you're going to decrement the starting point of the indices, then you need to increment that index wherever it appears in the summand. 2) Typically, the coefficients in a generating function will start at $n=0$. That way, one has $g(0)=a_0$, $g'(0)=a_1$, $g''(0)=2a_2$, and so forth.
– Semiclassical Aug 19 '21 at 03:31 -
1$n$ can start at any integer (positive, zero or even negative numbers). The rule of thumb is chose one which make the algebra simpler and hence your life easier. – achille hui Aug 19 '21 at 06:26