A114115 -id:A114115 - cp's OEIS frontend

A114116 1's-counting matrix: row sums give number of 1's in binary expansion of n+1.

1, 0, 1, 2, -1, 1, -1, 2, -1, 1, 1, 0, 1, -1, 1, 1, 0, 0, 1, -1, 1, 3, -2, 2, -1, 1, -1, 1, -2, 3, -2, 2, -1, 1, -1, 1, 0, 1, 0, 0, 1, -1, 1, -1, 1, 0, 1, 0, 0, 0, 1, -1, 1, -1, 1, 2, -1, 2, -2, 2, -1, 1, -1, 1, -1, 1, 0, 1, -1, 2, -2, 2, -1, 1, -1, 1, -1, 1, 2, -1, 1, 0, 0, 0, 1, -1, 1, -1, 1, -1, 1, 2, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 1, -1, 1, 4, -3, 3, -2, 2

Offset: 0

Paul Barry, Nov 13 2005

First column is -A037861(n+1). Row sums are A000120. Product of partial sum matrix (1/(1-x),x) and A114115. Inverse is A114117.

			Triangle begins
1;
0, 1;
2,-1, 1;
-1, 2,-1, 1;
1, 0, 1,-1, 1;
1, 0, 0, 1,-1, 1;
3,-2, 2,-1, 1,-1, 1;

A114117 Inverse of 1's counting matrix A114116.

Original entry on oeis.org

1, 0, 1, -2, 1, 1, -1, -1, 1, 1, 0, -2, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, 0, -2, 0, 0, 1, 1, 0, 0, -1, -1, 0, 0, 1, 1, 0, 0, 0, -2, 0, 0, 0, 1, 1, 0, 0, 0, -1, -1, 0, 0, 0, 1, 1, 0, 0, 0, 0, -2, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 0

Paul Barry, Nov 13 2005

Row sums are (1,1,0,0,0,.....) with g.f. 1+x. Diagonal sums have g.f. (1-x^2-x^3)/(1-x^3). Product of A114115 and the first difference matrix (1-x,x).

			Triangle begins
  1;
  0, 1;
 -2, 1, 1;
 -1,-1, 1, 1;
  0,-2, 0, 1, 1;
  0,-1,-1, 0, 1, 1;
  0, 0,-2, 0, 0, 1, 1;
  0, 0,-1,-1, 0, 0, 1, 1;

Cf. A114115, A114116.

T(n, k) = Sum_{j=0..n} Sum_{i=0..n} C(floor((n+i)/2), j)*C(j, floor((n+i)/2))*(2*C(0, j-k)-C(1, j-k)).

A114114 An invertible partition matrix.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 0, 1, 2, 1, 0, 0, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 0, 2, 2, 2, 1, 0, 0, 0, 1, 2, 2, 2, 1, 0, 0, 0, 0, 2, 2, 2, 2, 1, 0, 0, 0, 0, 1, 2, 2, 2, 2, 1, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 1, 0, 0, 0, 0, 0, 1, 2, 2, 2, 2, 2, 1, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 1

Offset: 0

Paul Barry, Nov 13 2005

Row sums are n+1, A000027. Diagonal sums are 1,1,1,2,2,2,3,3,3,.... or A008620. Inverse is A114115. Product with first difference matrix (1-x,x) is A114117.

			Triangle begins
1.................=1
1,1...............=2
0,2,1.............=3
0,1,2,1...........=4
0,0,2,2,1.........=5
0,0,1,2,2,1.......=6
0,0,0,2,2,2,1.....=7
0,0,0,1,2,2,2,1...=8

Number triangle T(n, k)=sum{j=0..n, C(floor((n+j)/2), k)C(k, floor((n+j)/2))}.

cp's OEIS Frontend

A114116 1's-counting matrix: row sums give number of 1's in binary expansion of n+1.

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A114117 Inverse of 1's counting matrix A114116.

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A114114 An invertible partition matrix.

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