A207974 Triangle related to A152198.

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

Author

Philippe Deléham, Feb 22 2012

Keywords

• easy
• nonn
• tabl

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

• Gheorghe Coserea, Rows n = 0..200, flattened

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