A182534 Array read by antidiagonals: coefficient of the Euler-Mascheroni constant in below expression.

1, 1, 2, 2, 2, 3, 5, 4, 2, 6, 14, 10, 3, 4, 10, 42, 28, 6, 6, 5, 20, 132, 84, 14, 12, 6, 10, 35, 429, 264, 36, 28, 10, 12, 14, 70, 1430, 858, 99, 72, 20, 20, 14, 28, 126, 4862, 2860, 286, 198, 45, 40, 20, 28, 42, 252

Offset

1,3

Comments

The (i,j)-entry of the array is the coefficient of the Euler-Mascheroni constant in: -2^(i+2j-1)/Pi*int(log(x)*cos(x)^i*sin(x)^(2j-1)/x, x=0..infinity); see Mathematica code below.

First row: A000108.

Second row: -A002420.

Third row: A007054.

Fourth row: A002421.

Fifth row: A007272.

Sixth row: -A002422.

Eighth row: A002423.

First column: A001405.

Second column: A089408.

Odd entries on main diagonal: A126596.

Links

Table of n, a(n) for n=1..55.

Example

Evaluate: -256/Pi*int(cos(x)^3*log(x)*sin(x)^5/x, x=0..infinity) = 3*eulergamma-log(9/8). Thus the (3,3) entry of the array is 3, the coefficient of the Euler-Mascheroni constant in this expression.
The array begins as:
| 1   1   2   5   14  42  132 429  ... |
| 2   2   4   10  28  84  264 858  ... |
| 3   2   3   6   14  36  99  286  ... |
| 6   4   6   12  28  72  198 572  ... |
| 10  5   6   10  20  45  110 286  ... |
| 20  10  12  20  40  90  220 572  ... |
| 35  14  14  20  35  70  154 364  ... |
| 70  28  28  40  70  140 308 728  ... |
| ... ... ... ... ... ... ... ...  ... |

Mathematica

A[a_, b_] :=
  A[a, b] =
   Array[Coefficient[
      Integrate[
        Log[x]*Cos[x]^#1*Sin[x]^(2 #2 - 1)/x, {x, 0,
         Infinity}] (2^(#1 + 2 #2 - 1))/(-\[Pi]), EulerGamma] &, {a, b}];
A[11, 11];
Print[A[11, 11] // MatrixForm];
Table2 = {};
k = 1;
While[k < 11, Table1 = {};
  i = 1;
  j = k;
  While[0 < j,
    AppendTo[Table1,
    First[Take[First[Take[A[11, 11], {i, i}]], {j, j}]]];
    j = j - 1;
    i = i + 1];
    AppendTo[Table2, Table1];
    k++];
Print[Flatten[Table2]]

Crossrefs

Cf. A000108, A002420, A007054, A002421, A007272, A002422, A002423, A001405, A089408, A126596.

Sequence in context: A160902 A082408 A270566 * A165918 A114639 A071867

Adjacent sequences: A182531 A182532 A182533 * A182535 A182536 A182537

Keyword

nonn,tabl

Author

John M. Campbell, May 05 2012

Status

approved