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MAPLE
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b:= proc(u, o, c) option remember; `if`(c<0 or c>7, 0, `if`(u+o=0,
x^c, (p-> add(coeff(p, x, i)*x^max(i, c), i=0..7))(add(
b(u-j, o-1+j, c-1), j=1..u)+add(b(u+j-1, o-j, c+1), j=1..o))))
end:
a:= n-> coeff(add(b(j-1, n-j, 0), j=1..n), x, 7):
seq(a(n), n=7..30);
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