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Array T(n,m) = A177944(2*n,2*m) read by antidiagonals.
1

%I #28 Oct 14 2024 01:35:39

%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