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

%I A182534 #19 Jan 30 2013 03:32:49
%S A182534 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 A182534 429,264,36,28,10,12,14,70,1430,858,99,72,20,20,14,28,126,4862,2860,
%U A182534 286,198,45,40,20,28,42,252
%N A182534 Array read by antidiagonals: coefficient of the Euler-Mascheroni constant in below expression.
%C A182534 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 A182534 First row:     A000108.
%C A182534 Second row:   -A002420.
%C A182534 Third row:     A007054.
%C A182534 Fourth row:    A002421.
%C A182534 Fifth row:     A007272.
%C A182534 Sixth row:    -A002422.
%C A182534 Eighth row:    A002423.
%C A182534 First column:  A001405.
%C A182534 Second column: A089408.
%C A182534 Odd entries on main diagonal: A126596.
%e A182534 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 A182534 The array begins as:
%e A182534 | 1   1   2   5   14  42  132 429  ... |
%e A182534 | 2   2   4   10  28  84  264 858  ... |
%e A182534 | 3   2   3   6   14  36  99  286  ... |
%e A182534 | 6   4   6   12  28  72  198 572  ... |
%e A182534 | 10  5   6   10  20  45  110 286  ... |
%e A182534 | 20  10  12  20  40  90  220 572  ... |
%e A182534 | 35  14  14  20  35  70  154 364  ... |
%e A182534 | 70  28  28  40  70  140 308 728  ... |
%e A182534 | ... ... ... ... ... ... ... ...  ... |
%t A182534 A[a_, b_] :=
%t A182534   A[a, b] =
%t A182534    Array[Coefficient[
%t A182534       Integrate[
%t A182534         Log[x]*Cos[x]^#1*Sin[x]^(2 #2 - 1)/x, {x, 0,
%t A182534          Infinity}] (2^(#1 + 2 #2 - 1))/(-\[Pi]), EulerGamma] &, {a, b}];
%t A182534 A[11, 11];
%t A182534 Print[A[11, 11] // MatrixForm];
%t A182534 Table2 = {};
%t A182534 k = 1;
%t A182534 While[k < 11, Table1 = {};
%t A182534   i = 1;
%t A182534   j = k;
%t A182534   While[0 < j,
%t A182534     AppendTo[Table1,
%t A182534     First[Take[First[Take[A[11, 11], {i, i}]], {j, j}]]];
%t A182534     j = j - 1;
%t A182534     i = i + 1];
%t A182534     AppendTo[Table2, Table1];
%t A182534     k++];
%t A182534 Print[Flatten[Table2]]
%Y A182534 Cf. A000108, A002420, A007054, A002421, A007272, A002422, A002423, A001405, A089408, A126596.
%K A182534 nonn,tabl
%O A182534 1,3
%A A182534 _John M. Campbell_, May 05 2012