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A177970 Array T(n,m) = A177944(2*n,2*m) read by antidiagonals.

%I A177970 #28 Oct 14 2024 01:35:39
%S A177970 1,1,1,1,26,1,1,99,99,1,1,244,622,244,1,1,485,2300,2300,485,1,1,846,
%T A177970 6423,12000,6423,846,1,1,1351,15001,45031,45031,15001,1351,1,1,2024,
%U A177970 30924,136120,218774,136120,30924,2024,1,1,2889,58122,352698,831384
%N A177970 Array T(n,m) = A177944(2*n,2*m) read by antidiagonals.
%C A177970 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 A177970 Robert Israel, <a href="/A177970/b177970.txt">Table of n, a(n) for n = 0..10000</a>
%F A177970 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 A177970 The table starts in row n=0, column m=0 as:
%e A177970  1,     1,      1,      1,       1,         1,         1,         1,
%e A177970  1,    26,     99,    244,     485,       846,      1351,      2024,
%e A177970  1,    99,    622,   2300,    6423,     15001,     30924,     58122,
%e A177970  1,   244,   2300,  12000,   45031,    136120,    352698,    813940,
%e A177970  1,   485,   6423,  45031,  218774,    831384,   2645350,   7354688,
%e A177970  1,   846,  15001, 136120,  831384,   3879856,  14872836,  49031376,
%e A177970  1,  1351,  30924, 352698, 2645350,  14872836,  67603876, 260757874,
%e A177970  1,  2024,  58122, 813940, 7354688,  49031376, 260757874,1163381372,
%p A177970 T:= (m,n) -> (2*n+1)*binomial(2*m+1+2*n, 2*m)-2*n-2*m:
%p A177970 seq(seq(T(m,s-m),m=0..s),s=0..10); # _Robert Israel_, Jul 06 2017
%t A177970 t[n_, m_] = 1/Beta[2*n + 1, 2*m + 1] - 2*n - 2*m;
%t A177970 a = Table[Table[t[n, m], {m, 0, 10}], {n, 0, 10}];
%t A177970 Table[Table[a[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}];
%t A177970 Flatten[%]
%o A177970 (Python)
%o A177970 from sympy import binomial
%o A177970 def T(m, n): return (2*n + 1)*binomial(2*m + 1 + 2*n, 2*m) - 2*n - 2*m
%o A177970 for n in range(11): print([T(m, n - m) for m in range(n + 1)]) # _Indranil Ghosh_, Jul 06 2017
%K A177970 nonn,easy,tabl,look
%O A177970 0,5
%A A177970 _Roger L. Bagula_, May 16 2010
%E A177970 Definition rewritten with A177944, examples brought into normal form, closed sum formula - The Assoc. Eds. of the OEIS, Nov 03 2010