cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

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).

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Aug 01 2010

Keywords

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

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

A207974 Triangle related to A152198.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 4, 2, 2, 1, 1, 5, 2, 4, 1, 1, 1, 6, 3, 6, 3, 2, 1, 1, 7, 3, 9, 3, 5, 1, 1, 1, 8, 4, 12, 6, 8, 4, 2, 1, 1, 9, 4, 16, 6, 14, 4, 6, 1, 1, 1, 10, 5, 20, 10, 20, 10, 10, 5, 2, 1
Offset: 0

Views

Author

Philippe Deléham, Feb 22 2012

Keywords

Comments

Row sums are A027383(n).
Diagonal sums are alternately A014739(n) and A001911(n+1).
The matrix inverse starts
1;
-1,1;
1,-2,1;
1,-1,-1,1;
-1,2,0,-2,1;
-1,1,2,-2,-1,1;
1,-2,-1,4,-1,-2,1;
1,-1,-3,3,3,-3,-1,1;
-1,2,2,-6,0,6,-2,-2,1;
-1,1,4,-4,-6,6,4,-4,-1,1;
1,-2,-3,8,2,-12,2,8,-3,-2,1;
apparently related to A158854. - R. J. Mathar, Apr 08 2013
From Gheorghe Coserea, Jun 11 2016: (Start)
T(n,k) is the number of terms of the sequence A057890 in the interval [2^n,2^(n+1)-1] having binary weight k+1.
T(n,k) = A007318(n,k) (mod 2) and the number of odd terms in row n of the triangle is 2^A000120(n).
(End)

Examples

			Triangle begins :
n\k  [0] [1] [2] [3] [4] [5] [6] [7] [8] [9]
[0]  1;
[1]  1,  1;
[2]  1,  2,  1;
[3]  1,  3,  1,  1;
[4]  1,  4,  2,  2,  1;
[5]  1,  5,  2,  4,  1,  1;
[6]  1,  6,  3,  6,  3,  2,  1;
[7]  1,  7,  3,  9,  3,  5,  1,  1;
[8]  1,  8,  4,  12, 6,  8,  4,  2,  1;
[9]  1,  9,  4,  16, 6,  14, 4,  6,  1,  1;
[10] ...

Links

Crossrefs

Cf. Columns : A000012, A000027, A004526, A002620, A008805, A006918, A058187
Cf. Diagonals : A000012, A000034, A052938, A097362
Cf. A007318, A110813, A135278, A152201
Related to thickness: A000120, A027383, A057890, A274036.

Programs

  • Maple
    A207974 := proc(n,k)
    if k = 0 then
        1;
    elif k < 0 or k > n then
        0 ;
    else
        procname(n-1,k-1)-(-1)^k*procname(n-1,k) ;
    end if;
end proc: # R. J. Mathar, Apr 08 2013
  • PARI
    seq(N) = {
  my(t = vector(N+1, n, vector(n, k, k==1 || k == n)));
  for(n = 2, N+1, for (k = 2, n-1,
      t[n][k] = t[n-1][k-1] + (-1)^(k%2)*t[n-1][k]));
  return(t);
};
concat(seq(10))  \\ Gheorghe Coserea, Jun 09 2016
  • PARI
    P(n) = ((2+x+(n%2)*x^2) * (1+x^2)^(n\2) - 2)/x;
concat(vector(11, n, Vecrev(P(n-1)))) \\ Gheorghe Coserea, Mar 14 2017

Formula

T(n,k) = T(n-1,k-1) - (-1)^k*T(n-1,k), k>0 ; T(n,0) = 1.
T(2n,2k) = T(2n+1,2k) = binomial(n,k) = A007318(n,k).
T(2n+1,2k+1) = A110813(n,k).
T(2n+2,2k+1) = 2*A135278(n,k).
T(n,2k) + T(n,2k+1) = A152201(n,k).
T(n,2k) = A152198(n,k).
T(n+1,2k+1) = A152201(n,k).
T(n,k) = T(n-2,k-2) + T(n-2,k).
T(2n,n) = A128014(n+1).
T(n,k) = card {p, 2^n <= A057890(p) <= 2^(n+1)-1 and A000120(A057890(p)) = k+1}. - Gheorghe Coserea, Jun 09 2016
P_n(x) = Sum_{k=0..n} T(n,k)*x^k = ((2+x+(n mod 2)*x^2)*(1+x^2)^(n\2) - 2)/x. - Gheorghe Coserea, Mar 14 2017
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.