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

0,5

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)

Links

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

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

Example

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] ...

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

Prog

(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

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.

Sequence in context: A105535 A182980 A244051 * A239454 A108888 A124021

Adjacent sequences: A207971 A207972 A207973 * A207975 A207976 A207977

Keyword

easy,nonn,tabl

Author

Philippe Deléham, Feb 22 2012

Status

approved