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MAPLE
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with(combinat):
T:= proc(n, j) option remember; binomial(n-1, j-1)*n^(n-j) end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
expand(add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
x^(igcd(i, 4)*j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> add((p-> add(n^i*T(n, j)* coeff(p, x, i),
i=0..degree(p)))(b(j$2)), j=0..n):
seq(a(n), n=0..20);
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