A169950 Consider the 2^n monic polynomials f(x) with coefficients 0 or 1 and degree n. Sequence gives triangle read by rows, in which T(n,k) (n>=0) is the number of such polynomials of thickness k (1 <= k <= n+1).

1, 1, 1, 1, 2, 1, 1, 5, 1, 1, 1, 8, 4, 2, 1, 1, 13, 8, 8, 1, 1, 1, 20, 15, 18, 7, 2, 1, 1, 33, 23, 45, 13, 11, 1, 1, 1, 48, 44, 86, 36, 28, 10, 2, 1, 1, 75, 64, 184, 70, 84, 18, 14, 1, 1, 1, 100, 117, 332, 166, 188, 68, 36, 13, 2, 1, 1, 145, 173, 657, 282, 482, 134, 132, 23, 17, 1, 1

Offset: 0

Author

N. J. A. Sloane, Aug 01 2010

Keywords

• nonn
• tabl

Comments

The thickness of a polynomial f(x) is the magnitude of the largest coefficient in the expansion of f(x)^2.

Examples

			Triangle begins:
n\k  [1]   [2]   [3]   [4]   [5]   [6]   [7]   [8]   [9]   [10]  [11]  [12]
[0]  1;
[1]  1,    1;
[2]  1,    2,    1;
[3]  1,    5,    1,    1;
[4]  1,    8,    4,    2,    1;
[5]  1,    13,   8,    8,    1,    1;
[6]  1,    20,   15,   18,   7,    2,    1;
[7]  1,    33,   23,   45,   13,   11,   1,    1;
[8]  1,    48,   44,   86,   36,   28,   10,   2,    1;
[9]  1,    75,   64,   184,  70,   84,   18,   14,   1,    1;
[10] 1,    100,  117,  332,  166,  188,  68,   36,   13,   2,    1;
[11] 1,    145,  173,  657,  282,  482,  134,  132,  23,   17,   1,    1;
[12] ...
For n = 3, the eight polynomials, their squares and thicknesses are as follows:
x^3, x^6, 1
x^3+1, x^6+2*x^3+1, 2
x^3+x, x^6+2*x^4+x^2, 2
x^3+x+1, x^6+2*x^4+2*x^3+x^2+2*x+1, 2
x^3+x^2, x^6+2*x^5+x^4, 2
x^3+x^2+1, x^6+2*x^5+2*x^3+x^4+2*x^2+1, 2
x^3+x^2+x, x^6+2*x^5+3*x^4+2*x^3+x^2, 3
x^3+x^2+x+1, x^6+2*x^5+3*x^4+4*x^3+3*x^2+2*x+1, 4
Hence T(3,1) = 1, T(3,2) = 5, T(3,3) = 1, T(3,4) = 1.
		

Links

• Gheorghe Coserea, Rows n = 0..33, flattened
• Index entries for sequences related to carryless arithmetic

Crossrefs

Related to thickness: A169940-A169954, A061909, A274036.

Programs

Mathematica

Last /@ Tally@ # & /@ Table[Max@ CoefficientList[SeriesData[x, 0, #, 0, 2^n, 1]^2, x] &@ IntegerDigits[#, 2] & /@ Range[2^n, 2^(n + 1) - 1], {n, 12}] // Flatten (* Michael De Vlieger, Jun 08 2016 *)

PARI

seq(n) = {
  my(a = vector(n+1, k, vector(k)), x='x);
  for(k = 1, 2^(n+1)-1, my(pol = Pol(binary(k), x));
       a[poldegree(pol)+1][vecmax(Vec(sqr(pol)))]++);
  return(a);
};
concat(seq(11))  \\ Gheorghe Coserea, Jun 06 2016

Formula

Wanted: a recurrence. Are any of A169940-A169954 related to any other entries in the OEIS?

Extensions

Rows 17-30 of the triangle from Nathaniel Johnston, Nov 15 2010