How to ask mathematica to compute higher order derivatives evaluated at 0 -

i have function, let's example,

d[x^2*exp[x^2], {x, 6}] /. x -> 0

and want replace 6 general integer n,

or cases following:

 limit[limit[d[d[x /((-1 + x) (1 - y) (-1 + x + x y)), {x, 3}], {y, 5}], {x -> 0}], {y -> 0}]

and want replace 3 , 5 general integer m , n respectively.

how solve these 2 kinds of problems in general in mma?

many thanks.

can use seriescoefficient, sometimes.

inputform[n! * seriescoefficient[x^2*exp[x^2], {x,0,n}]]

out[21]//inputform= n!*piecewise[{{gamma[n/2]^(-1), mod[n, 2] == 0 && n >= 2}}, 0]

inputform[mncoeff = m!*n! *   seriescoefficient[x/((-1+x)*(1-y)*(-1+x+x*y)), {x,0,m}, {y,0,n}]]

out[22]//inputform= m!*n!*piecewise[{{-1 + binomial[m, 1 + n]*hypergeometric2f1[1, -1 - n, m - n, -1], m >= 1 && n > -1}}, 0]

good luck extracting limits m, n integer, in second case.

daniel lichtblau wolfram research

Assebmley