A128139 Triangle read by rows: matrix product A004736 * A128132.

1, 1, 2, 1, 3, 3, 1, 4, 5, 4, 1, 5, 7, 7, 5, 1, 6, 9, 10, 9, 6, 1, 7, 11, 13, 13, 11, 7, 1, 8, 13, 16, 17, 16, 13, 8, 1, 9, 15, 19, 21, 21, 19, 15, 9, 1, 10, 17, 22, 25, 26, 25, 22, 17, 10

Offset: 0

Author

Gary W. Adamson, Feb 16 2007

Keywords

• nonn
• easy
• tabl

Comments

A077028 with the final term in each row omitted.

Interchanging the factors in the matrix product leads to A128140 = A128132 * A004736.

From Gary W. Adamson, Jul 01 2012: (Start)

Alternatively, antidiagonals of an array A(n,k) of sequences with arithmetic progressions as follows:

1, 2, 3, 4, 5, 6, ...

1, 3, 5, 7, 9, 11, ...

1, 4, 7, 10, 13, 16, ...

1, 5, 9, 13, 17, 21, ...

... (End)

From Gary W. Adamson, Jul 02 2012: (Start)

A summation generalization for Sum_{k>=1} 1/(A(n,k)*A(n,k+1)) (formulas copied from A002378, A000466, A085001, A003185):

1 = 1/(1)*(2) + 1/(2)*(3) + 1/(3)*(4) + ...

1 = 2/(1)*(3) + 2/(3)*(5) + 2/(5)*(7) + ...

1 = 3/(1)*(4) + 3/(4)*(7) + 3/(7)*(10) + ...

1 = 4/(1)*(5) + 4/(5)*(9) + 4/(9)*(13) + ...

...

As a summation of terms equating to a definite integral:

Integral_{0..1} dx/(1+x) = ... 1 - 1/2 + 1/3 - 1/4 + ... = log(2).

Integral_{0..1} dx/(1+x^2) = 1 - 1/3 + 1/5 - 1/7 + ... = Pi/4 (see A157142)

Integral_{0..1} dx/(1+x^3) = 1 - 1/4 + 1/7 - 1/10 + ... (see A016777)

Integral_{0..1} dx/(1+x^4) = 1 - 1/5 + 1/9 - 1/13 + ... (see A016813). (End)

Examples

			First few rows of the triangle:
  1;
  1,  2;
  1,  3,  3;
  1,  4,  5,  4;
  1,  5,  7,  7,  5;
  1,  6,  9, 10,  9,  6;
  1,  7, 11, 13, 13, 11,  7;
  1,  8, 13, 16, 17, 16, 13,  8;
  1,  9, 15, 19, 21, 21, 19, 15,  9;
  1, 10, 17, 22, 25, 26, 25, 22, 17, 10;
  ...
		

Crossrefs

Cf. A004736, A128132, A128140, A004006 (row sums).

Formula

A004736 * A128132 as infinite lower triangular matrices.

T(n,k) = k*(1+n-k)+1 = 1 + A094053(n+1,1+n-k). - R. J. Mathar, Jul 09 2012