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%I #23 May 20 2021 08:19:30
%S 1,1,1,1,26,1,1,99,99,1,1,244,622,244,1,1,485,2300,2300,485,1,1,846,
%T 6423,12000,6423,846,1,1,1351,15001,45031,45031,15001,1351,1,1,2024,
%U 30924,136120,218774,136120,30924,2024,1,1,2889,58122,352698,831384
%N Array T(n,m) = A177944(2*n,2*m) read by antidiagonals.
%C Antidiagonal sums are 1, 2, 28, 200, 1112, 5572, 26540, 122768, 556912, 2490188, ... = 4^d*(d+1/2)-2*d(d+1), d > 0.
%H Robert Israel, <a href="/A177970/b177970.txt">Table of n, a(n) for n = 0..10000</a>
%F T(n,m) = 1/Beta(2*n+1, 2*m+1) - 2*n - 2*m where Beta(a,b) = Gamma(a)*Gamma(b)/(Gamma(a+b) .
%e The table starts in row n=0, column m=0 as:
%e .1,.....1,......1,......1,.......1,.........1,.........1,.........1,
%e .1,....26,.....99,....244,.....485,.......846,......1351,......2024,
%e .1,....99,....622,...2300,....6423,.....15001,.....30924,.....58122,
%e .1,...244,...2300,..12000,...45031,....136120,....352698,....813940,
%e .1,...485,...6423,..45031,..218774,....831384,...2645350,...7354688,
%e .1,...846,..15001,.136120,..831384,...3879856,..14872836,..49031376,
%e .1,..1351,..30924,.352698,.2645350,..14872836,..67603876,.260757874,
%e .1,..2024,..58122,.813940,.7354688,..49031376,.260757874,1163381372,
%p T:= (m,n) -> (2*n+1)*binomial(2*m+1+2*n, 2*m)-2*n-2*m:
%p seq(seq(T(m,s-m),m=0..s),s=0..10); # _Robert Israel_, Jul 06 2017
%t t[n_, m_] = 1/Beta[2*n + 1, 2*m + 1] - 2*n - 2*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[%]
%o (Python)
%o from sympy import binomial
%o def T(m, n): return (2*n + 1)*binomial(2*m + 1 + 2*n, 2*m) - 2*n - 2*m
%o for n in range(11): print([T(m, n - m) for m in range(n + 1)]) # _Indranil Ghosh_, Jul 06 2017
%K nonn,easy,tabl,look
%O 0,5
%A _Roger L. Bagula_, May 16 2010
%E Definition rewritten with A177944, examples brought into normal form, closed sum formula - The Assoc. Eds. of the OEIS, Nov 03 2010
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