A346379 The least k such that the number of real roots of Sum_{i=0..j} (i+1)^k*x^i is j for j = 0,1,...,n or -1 if no solution for this n exists.

0, 0, 5, 12, 21, 33, 47, 63, 81, 102, 125, 151, 179, 209, 242, 277, 314, 354, 396, 440, 487, 536, 587, 641, 697, 755, 816, 879, 944, 1012, 1082, 1155, 1230, 1307, 1387, 1468, 1553, 1639, 1728, 1819, 1913, 2009, 2107, 2208, 2311, 2416, 2524, 2634, 2747, 2861, 2978

Offset

0,3

Comments

Conjecture: The sequence is defined for all n.

A Maple calculation shows that for n = 125 and k = 20000 the number of real roots of Sum_{i=0..j} (i+1)^k*x^i is j for j = 0,1,...,n. This shows that a(n) is defined at least up to n = 125.

Links

Table of n, a(n) for n=0..50.

W. Edwin Clark and Mark Shattuck, 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, arXiv:2107.05572 [math.CO], 2021.

Maple

Section:=proc(p, j)
local  i, x;
x:=op(indets(p));
add(coeff(p, x, i)*x^i, i=0..j);
end:
IsGood:=proc(p)
local i, n, x;
n:=degree(p);
for i from 0 to n do
  if NumRealRoots(Section(p, i)) <> i then return false; fi;
od:
return true;
end:
NumRealRoots:=proc(p)
local q, k, u;
if p = 0 then error "zero polynomial not allowed"; fi;
q:=sqrfree(p);
  k:=0;
  for u in q[2] do
    k:=k+nops(realroot(u[1]))*u[2];
  od;
  k;
end:
K0:=0:
for n from 0 to 50 do
   if n = 0 then printf("%d, ", 0); next;  fi;
flag:=false;
   for k from K0 to 10^10 do
  p:=add(((i+1)^k)*x^i, i=0..n);
    if IsGood(p) then  K0:=k; printf("%d, ", k); flag:=true;        break; fi;
   od:
if flag=false then printf("%d, ", -1); fi;
od:

Prog

(PARI) f(n, k) = (n+1)^k;
nb(n, k) = {my(v = vector(n+1, i, f(i-1, k))); #polrootsreal(Pol(v)); }
a(n) = {my(k=0); while (vector(n+1, i, nb(i-1, k)) != [0..n], k++); k; } \\ Michel Marcus, Jul 14 2021

Crossrefs

Sequence in context: A272451 A097984 A028347 * A354399 A256320 A301693

Adjacent sequences: A346376 A346377 A346378 * A346380 A346381 A346382

Keyword

nonn,hard

Author

W. Edwin Clark, Jul 14 2021

Extensions

a(29)-a(43) from Hugo Pfoertner, Jul 17 2021

a(44)-a(50) from Hugo Pfoertner, Jul 19 2021

Status

approved