read HISTABRUT:
print(`Consider the random variable: `):
print(`area under a lattice walk in the plane from the origin to (n,n) never going`):
print(`above the diagonal (given by the q-Catalan numbers `):
gu:= Yofseq([seq(normal(qbin(2*i,i,q)/(1-q^(i+1))*(1-q)*(i+1)),i=1..30)],q,12,n,24,1,r) :
print(`The mean is:`, gu[1]):
print(`The variance is:` , gu[2]):
print(`The asympotics to order 1, of the even alpha coefficients (the (2r)-th moment about the mean divided by the r-th power of the variance) `):
print(`as an expression in n and r is:`):
print(gu[3]):
print(`the (normalized) odd moments, to order 1, are`):
print(gu[4]):
print(``):
print(`In particular it is asymptotically normal, but we have an even finer asympotics for the alpha coefficients. `):
print(`This took `, time(), `seconds . `):
quit: