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A336521 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is the coefficient of x^(k*n) in expansion of ( (1 + x)/(1 - x) )^n.

%I A336521 #38 Aug 09 2025 20:50:56
%S A336521 1,1,1,1,2,1,1,2,8,1,1,2,16,38,1,1,2,24,146,192,1,1,2,32,326,1408,
%T A336521 1002,1,1,2,40,578,4672,14002,5336,1,1,2,48,902,11008,69002,142000,
%U A336521 28814,1,1,2,56,1298,21440,216002,1038984,1459810,157184,1,1,2,64,1766,36992,525002,4320608,15856206,15158272,864146,1
%N A336521 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is the coefficient of x^(k*n) in expansion of ( (1 + x)/(1 - x) )^n.
%H A336521 Seiichi Manyama, <a href="/A336521/b336521.txt">Antidiagonals n = 0..139, flattened</a>
%F A336521 T(n,k) = (1/k) * [x^n] ( (1 + x)/(1 - x) )^(k*n) for k > 0 and n > 0.
%F A336521 T(n,k) = Sum_{j=0..n} binomial(n,j) * binomial(k*n+j-1,n-1).
%F A336521 T(n,k) = (1/k) * Sum_{j=0..n} binomial(k*n,n-j) * binomial(k*n+j-1,j) for k > 0 and n > 0.
%F A336521 T(n,k) = Sum_{j=1..n} 2^j * binomial(n,j) * binomial(k*n-1,j-1) for n > 0.
%F A336521 T(n,k) = binomial(k*n-1, n-1)*hypergeom([-n, k*n], [1+(k-1)*n], -1) for k > 0. - _Stefano Spezia_, Aug 09 2025
%e A336521 Square array begins:
%e A336521   1,    1,     1,     1,      1,      1, ...
%e A336521   1,    2,     2,     2,      2,      2, ...
%e A336521   1,    8,    16,    24,     32,     40, ...
%e A336521   1,   38,   146,   326,    578,    902, ...
%e A336521   1,  192,  1408,  4672,  11008,  21440, ...
%e A336521   1, 1002, 14002, 69002, 216002, 525002, ...
%t A336521 T[n_, 0] := 1; T[n_, k_] := Sum[Binomial[n, j] * Binomial[k*n + j - 1, n - 1], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Jul 24 2020 *)
%Y A336521 Column k=0-3 give A000012, A123164, A103885, A333715.
%Y A336521 Main diagonal gives A336522.
%Y A336521 Cf. A122542, A266213, A336534.
%K A336521 nonn,tabl
%O A336521 0,5
%A A336521 _Seiichi Manyama_, Jul 24 2020