|
|
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).
|
|
7
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,5
|
|
COMMENTS
|
The thickness of a polynomial f(x) is the magnitude of the largest coefficient in the expansion of f(x)^2.
|
|
LINKS
|
|
|
FORMULA
|
Wanted: a recurrence. Are any of A169940-A169954 related to any other entries in the OEIS?
|
|
EXAMPLE
|
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.
|
|
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 *)
|
|
PROG
|
(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);
};
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|