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%I #10 Aug 27 2019 08:19:15
%S 1,1,1,2,1,1,3,3,1,1,4,8,2,1,1,5,16,6,3,1,1,6,29,14,11,2,1,1,7,49,29,
%T 29,9,3,1,1,8,82,52,74,22,14,2,1,1,9,130,96,160,58,42,12,3,1,1,10,205,
%U 160,344,128,126,30,17,2,1,1,11,305,277,676,294,314,98,53,15,3,1
%N Consider the 2^(n+1) 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 (0 <= 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 <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 [1, 1]
%e [1, 2, 1]
%e [1, 3, 3, 1]
%e [1, 4, 8, 2, 1]
%e [1, 5, 16, 6, 3, 1]
%e [1, 6, 29, 14, 11, 2, 1]
%e [1, 7, 49, 29, 29, 9, 3, 1]
%e [1, 8, 82, 52, 74, 22, 14, 2, 1]
%e [1, 9, 130, 96, 160, 58, 42, 12, 3, 1]
%e [1, 10, 205, 160, 344, 128, 126, 30, 17, 2, 1]
%e [1, 11, 305, 277, 676, 294, 314, 98, 53, 15, 3, 1]
%e [1, 12, 450, 450, 1333, 576, 796, 232, 185, 38, 20, 2, 1]
%e [1, 13, 654, 712, 2477, 1177, 1796, 628, 501, 147, 64, 18, 3, 1]
%e [1, 14, 947, 1086, 4563, 2212, 4075, 1370, 1425, 368, 251, 46, 23, 2, 1]
%e [1, 15, 1343, 1657, 7997, 4289, 8535, 3265, 3515, 1117, 729, 205, 75, 21, 3, 1]
%e ...
%e For n=2 the eight polynomials and their squares are:
%e 0, 0
%e 1, 1
%e x, x^2
%e x+1, x^2+2*x+1
%e x^2, x^4
%e x^2+1, x^4+2*x^2+1
%e x^2+x, x^4+2*x^3+x^2
%e x^2+x+1, x^4+2*x^3+3*x^2+2*x+1
%e Their thicknesses are respectively 0,1,1,2,1,2,2,3, so T(2,0)=1, T(2,1)=T(2,2)=3, T(2,3)=1.
%e Contribution from R. J. Mathar, Aug 02 2010: (Start)
%e 1 16 1902 2448 14044 7686 17976 6728 8918 2586 2290 532 326 54 26 2 1
%e 1 17 2648 3636 23784 13932 35810 14690 20444 6928 6008 1798 990 272 86 24 3 1
%e 1 18 3675 5280 40196 24032 71374 29056 47604 15326 16590 4380 3384 724 410 62 29 2 1
%e 1 19 5015 7635 66066 42072 135876 59854 103298 37122 40660 12868 9346 2690 1290 348 97 27 3 1
%e 1 20 6824 10840 108419 70782 258898 114262 226282 78760 102664 29858 27848 6814 4752 944 503 70 32 2 1
%e 1 21 9166 15392 173675 120003 476494 224180 469816 178278 237956 78854 71902 21692 13698 3824 1625 433 108 30 3 1
%e 1 22 12343 21372 278183 196784 877315 414774 981244 364690 562404 176422 194486 52356 43592 9996 6407 1192 605 78 35 2 1 (End)
%t pols[n_] := x^Range[0, n].#& /@ Tuples[{0, 1}, n+1];
%t row[n_] := Tally[Max /@ CoefficientList[pols[n]^2, x]][[All, 2]];
%t Table[row[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Aug 27 2019 *)
%Y Related to thickness: A169940-A169954, A061909.
%Y 3rd column gives A143823(n+1)-n-2. - Alois P. Heinz, Sep 16 2011
%K nonn,tabf
%O 0,4
%A _N. J. A. Sloane_, Aug 01 2010
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