A182534 - OEIS

%I #19 Jan 30 2013 03:32:49

%S 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,

%T 429,264,36,28,10,12,14,70,1430,858,99,72,20,20,14,28,126,4862,2860,

%U 286,198,45,40,20,28,42,252

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

%C 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.

%C First row:     A000108.

%C Second row:   -A002420.

%C Third row:     A007054.

%C Fourth row:    A002421.

%C Fifth row:     A007272.

%C Sixth row:    -A002422.

%C Eighth row:    A002423.

%C First column:  A001405.

%C Second column: A089408.

%C Odd entries on main diagonal: A126596.

%e 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.

%e The array begins as:

%e | 1   1   2   5   14  42  132 429  ... |

%e | 2   2   4   10  28  84  264 858  ... |

%e | 3   2   3   6   14  36  99  286  ... |

%e | 6   4   6   12  28  72  198 572  ... |

%e | 10  5   6   10  20  45  110 286  ... |

%e | 20  10  12  20  40  90  220 572  ... |

%e | 35  14  14  20  35  70  154 364  ... |

%e | 70  28  28  40  70  140 308 728  ... |

%e | ... ... ... ... ... ... ... ...  ... |

%t A[a_, b_] :=

%t   A[a, b] =

%t    Array[Coefficient[

%t       Integrate[

%t         Log[x]*Cos[x]^#1*Sin[x]^(2 #2 - 1)/x, {x, 0,

%t          Infinity}] (2^(#1 + 2 #2 - 1))/(-\[Pi]), EulerGamma] &, {a, b}];

%t A[11, 11];

%t Print[A[11, 11] // MatrixForm];

%t Table2 = {};

%t k = 1;

%t While[k < 11, Table1 = {};

%t   i = 1;

%t   j = k;

%t   While[0 < j,

%t     AppendTo[Table1,

%t     First[Take[First[Take[A[11, 11], {i, i}]], {j, j}]]];

%t     j = j - 1;

%t     i = i + 1];

%t     AppendTo[Table2, Table1];

%t     k++];

%t Print[Flatten[Table2]]

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

%K nonn,tabl

%O 1,3

%A _John M. Campbell_, May 05 2012