%I #5 Dec 10 2016 17:17:49
%S 1,1,1,1,4,1,1,9,9,1,1,16,26,16,1,1,25,55,55,25,1,1,36,99,134,99,36,1,
%T 1,49,161,273,273,161,49,1,1,64,244,496,622,496,244,64,1,1,81,351,831,
%U 1251,1251,831,351,81,1
%N Array T(n,m) = 1/Beta(n+1, m+1) - n - m read by antidiagonals.
%C Antidiagonal sums are 1, 2, 6, 20, 60, 162, 406, 968, 2232, 5030, ... = (d+1)*(2^d-d).
%C The values appear to be related to binomial(n,m)^2.
%F T(n,m) = Gamma(n+m+2)/(Gamma(n+1)*Gamma(m+1)) - n - m = T(m,n).
%e The array starts in row n=0, column m=0 as:
%e 1,....1,....1,....1,.....1,.....1,.....1,.....1,
%e 1,....4,....9,...16,....25,....36,....49,....64, A000290
%e 1,....9,...26,...55,....99,...161,...244,...351, A154560
%e 1,...16,...55,..134,...273,...496,...831,..1310,
%e 1,...25,...99,..273,...622,..1251,..2300,..3949,
%e 1,...36,..161,..496,..1251,..2762,..5533,.10284,
%e 1,...49,..244,..831,..2300,..5533,.12000,.24011,
%e 1,...64,..351,.1310,..3949,.10284,.24011,.51466,
%t Clear[t, n];
%t t[n_, m_] = 1/Beta[n + 1, m + 1] - n - m;
%t a = Table[Table[t[n, m], {m, 0, 10}], {n, 0, 10}];
%t Table[Table[a[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}];
%t Flatten[%]
%K nonn,easy,tabl
%O 0,5
%A _Roger L. Bagula_, May 15 2010
%E Examples written in natural order, closed formula for antidiag. sum - The Assoc. Eds. of the OEIS, Nov 03 2010