The function powcreate creates a formal power series whose coefficients are specified by eqns.

The expression sequence of equations passed to powcreate should consist of one equation, possibly recursive, that specifies the nth coefficient, followed by zero or more equations that specify initial conditions.

More specifically, powcreate(t(n)=expr, t(0)=a0, ..., t(m)=am) creates a formal power series, t, whose nth coefficient is expr, and whose initial terms are $t\left(0\right)=\mathrm{a0}$, ..., $t\left(m\right)=\mathrm{am}$.

Any initial conditions among eqns override the general expression.  Thus, it is possible to include particular values of coefficients in eqns.

The command with(powseries,powcreate) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{powseries}\right)\:$

$\mathrm{powcreate}\left(t\left(n\right)=\frac{1}{n!}\,t\left(0\right)=1\right)\:$

$\mathrm{tpsform}\left(t\,x\,7\right)$

$1+x+\frac{1}{2}{x}^2+\frac{1}{6}{x}^3+\frac{1}{24}{x}^4+\frac{1}{120}{x}^5+\frac{1}{720}{x}^6+O\left(x^7\right)$

$\mathrm{powcreate}\left(v\left(n\right)=\frac{v\left(n-1\right)+v\left(n-2\right)}{4}\,v\left(0\right)=4\,v\left(1\right)=2\right)\:$

$\mathrm{tpsform}\left(v\,x\right)$

$4+2x+\frac{3}{2}{x}^2+\frac{7}{8}{x}^3+\frac{19}{32}{x}^4+\frac{47}{128}{x}^5+O\left(x^6\right)$