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%I A333988 #25 Sep 05 2020 03:13:01
%S A333988 1,1,1,1,2,1,1,3,8,1,1,4,17,32,1,1,5,28,99,128,1,1,6,41,208,577,512,1,
%T A333988 1,7,56,365,1552,3363,2048,1,1,8,73,576,3281,11584,19601,8192,1,1,9,
%U A333988 92,847,6016,29525,86464,114243,32768,1,1,10,113,1184,10033,62976,265721,645376,665857,131072,1
%N A333988 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of (1-(k+1)*x) / (1-2*(k+1)*x+((k-1)*x)^2).
%H A333988 Seiichi Manyama, <a href="/A333988/b333988.txt">Antidiagonals n = 0..139, flattened</a>
%F A333988 T(n,k) = Sum_{j=0..n} k^j * binomial(2*n,2*j).
%F A333988 T(0,k) = 1, T(1,k) = k+1 and T(n,k) = 2 * (k+1) * T(n-1,k) - (k-1)^2 * T(n-2,k) for n>1.
%e A333988 Square array begins:
%e A333988   1,   1,    1,     1,     1,     1, ...
%e A333988   1,   2,    3,     4,     5,     6, ...
%e A333988   1,   8,   17,    28,    41,    56, ...
%e A333988   1,  32,   99,   208,   365,   576, ...
%e A333988   1, 128,  577,  1552,  3281,  6016, ...
%e A333988   1, 512, 3363, 11584, 29525, 62976, ...
%t A333988 T[n_, 0] := 1; T[n_, k_] := Sum[k^j * Binomial[2*n, 2*j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Sep 04 2020 *)
%o A333988 (PARI) {T(n, k) = sum(j=0, n, k^j*binomial(2*n, 2*j))}
%Y A333988 Column k=0..9 give A000012, A081294, A001541, A090965, A083884, A099140, A099141, A099142, A165224, A026244.
%Y A333988 Main diagonal gives A333990.
%Y A333988 Cf. A009999, A307883, A337389, A333989.
%K A333988 nonn,tabl
%O A333988 0,5
%A A333988 _Seiichi Manyama_, Sep 04 2020

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