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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).
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%I #22 Jun 09 2016 03:28:43

%S 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,

%T 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,

%U 100,117,332,166,188,68,36,13,2,1,1,145,173,657,282,482,134,132,23,17,1,1

%N 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).

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

%H Gheorghe Coserea, <a href="/A169950/b169950.txt">Rows n = 0..33, flattened</a>

%H <a href="/index/Ca#CARRYLESS">Index entries for sequences related to carryless arithmetic</a>

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

%e Triangle begins:

%e n\k [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

%e [0] 1;

%e [1] 1, 1;

%e [2] 1, 2, 1;

%e [3] 1, 5, 1, 1;

%e [4] 1, 8, 4, 2, 1;

%e [5] 1, 13, 8, 8, 1, 1;

%e [6] 1, 20, 15, 18, 7, 2, 1;

%e [7] 1, 33, 23, 45, 13, 11, 1, 1;

%e [8] 1, 48, 44, 86, 36, 28, 10, 2, 1;

%e [9] 1, 75, 64, 184, 70, 84, 18, 14, 1, 1;

%e [10] 1, 100, 117, 332, 166, 188, 68, 36, 13, 2, 1;

%e [11] 1, 145, 173, 657, 282, 482, 134, 132, 23, 17, 1, 1;

%e [12] ...

%e For n = 3, the eight polynomials, their squares and thicknesses are as follows:

%e x^3, x^6, 1

%e x^3+1, x^6+2*x^3+1, 2

%e x^3+x, x^6+2*x^4+x^2, 2

%e x^3+x+1, x^6+2*x^4+2*x^3+x^2+2*x+1, 2

%e x^3+x^2, x^6+2*x^5+x^4, 2

%e x^3+x^2+1, x^6+2*x^5+2*x^3+x^4+2*x^2+1, 2

%e x^3+x^2+x, x^6+2*x^5+3*x^4+2*x^3+x^2, 3

%e x^3+x^2+x+1, x^6+2*x^5+3*x^4+4*x^3+3*x^2+2*x+1, 4

%e Hence T(3,1) = 1, T(3,2) = 5, T(3,3) = 1, T(3,4) = 1.

%t 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 *)

%o (PARI)

%o seq(n) = {

%o my(a = vector(n+1, k, vector(k)), x='x);

%o for(k = 1, 2^(n+1)-1, my(pol = Pol(binary(k), x));

%o a[poldegree(pol)+1][vecmax(Vec(sqr(pol)))]++);

%o return(a);

%o };

%o concat(seq(11)) \\ _Gheorghe Coserea_, Jun 06 2016

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

%K nonn,tabl

%O 0,5

%A _N. J. A. Sloane_, Aug 01 2010

%E Rows 17-30 of the triangle from _Nathaniel Johnston_, Nov 15 2010