Want a spline with the following conditions
f(0)=0    f(1)=1
f'(0)=0   f'(1)=0
f''(0)=0  f''(1)=rho

results in
y=ax5+bx4+cx^3

a=6+rho/2
b=-15-rho
c=10+rho/2

rho defines the curvature at x=1.

Usable range of rho is +infinity to -20
Beyond x<-20, the spline dips below zero.
Beyond x>0, the spline goes above 1.

theoretically interesting value of rho are -7.5 and -12 where y/x^2
quadratic undergoes interesting changes.

at -20 (the sharpest) we get

y=-4x5+5x4

The an S-shaped curve with no degrees of freedom

f(0)=0    f(1)=1
f'(0)=0   f'(1)=0

is

-2t3+3t2