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A207974 Triangle related to A152198. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified December 15 17:03 EST 2019. Contains 330000 sequences. (Running on oeis4.)