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We discuss the integer sequence transform  a 1→ b , where  bn  is the number of real roots of the polynomial  a0  +  a1x  +  a2x2  +  · · ·  +  a n x n. It is shown that several sequences  a  give the trivial sequence  b  = (0, 1, 0, 1, 0, 1, . . .), i.e.,  b n=  n  mod 2, among them the Catalan numbers, central binomial coefficients,  n ! and    for a fixed  k . We also look at some sequences  a  for which  b  is more interesting such as  a n= ( n  + 1)kfor  k ≥  3. Further, general procedures are given for constructing real sequences  a nfor which  b nis either always maximal or minimal.

quaestiones mathematicae

The integer sequence transform  a  →  b , where  b n                   is the number of real roots of the polynomial  a                  0  +  a                  1                  x  +  a                  2                  x                  2  +  · · ·  +  a n x n