Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #23 Feb 27 2022 16:11:27
%S 1,1,7,3,31,3,127,85,511,93,2047,105,8191,5461,32767,3855,131071,1533,
%T 524287,69905,299593,182361,8388607,1118481,33554431,22369621,
%U 19173961,9256395,536870911,11545611,2147483647,1431655765,8589934591
%N Reduced numerators of 2*(2^(1+n)-1)/(1+n)/(2+n).
%C a(m) is a numerator of the highest power of n coefficient in the sum of all matrix elements of n X n matrix M(i,j) = (i+j-1)^m, i,j=(1..n). E.g., a(5) = 3 because Sum_{j=1..n} Sum_{i=1..n} (i+j-1)^5 = (1/2)*(6n^7 - 5n^5 + n^3), a(6) = 127 because Sum_{j=1..n} Sum_{i=1..n} (i+j-1)^6 = (1/84)*n^2*(381n^6 - 434 n^4 + 147n^2 - 10). - _Alexander Adamchuk_, Apr 21 2006
%C a(n) is the numerator of Integral_{x=0..2} x^n*(1-abs(1-x)) dx. - _Groux Roland_, Jan 13 2011
%H J. Singh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Singh/singh8.html">On an Arithmetic Convolution</a>, J. Int. Seq. 17 (2014) # 14.6.7.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AbsoluteValue.html">Absolute Value</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnitSquareIntegral.html">Unit Square Integral</a>
%e 1, 1, 7/6, 3/2, 31/15, 3, 127/28, 85/12, 511/45, 93/5, 2047/66, ...
%t Table[(2(2^(n+1)-1))/((n+1)(n+2)),{n,0,40}]//Numerator (* _Harvey P. Dale_, Jul 14 2019 *)
%Y Cf. A116420.
%K nonn,frac
%O 0,3
%A _Eric W. Weisstein_, Feb 14 2006