%I #30 Feb 02 2021 17:11:16
%S 1,1,1,1,3,1,1,5,11,1,1,7,33,45,1,1,9,67,245,195,1,1,11,113,721,1921,
%T 873,1,1,13,171,1593,8179,15525,3989,1,1,15,241,2981,23649,95557,
%U 127905,18483,1,1,17,323,5005,54691,361449,1137709,1067925,86515,1
%N Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j*binomial(n,j)*binomial(2*j,j).
%H Seiichi Manyama, <a href="/A340970/b340970.txt">Antidiagonals n = 0..139, flattened</a>
%F G.f. of column k: 1/sqrt((1 - x) * (1 - (4*k+1)*x)).
%F T(n,k) = [x^n] (1+(2*k+1)*x+(k*x)^2)^n.
%F n * T(n,k) = (2*k+1) * (2*n-1) * T(n-1,k) - (4*k+1) * (n-1) * T(n-2,k) for n > 1.
%F E.g.f. of column k: exp((2*k+1)*x) * BesselI(0,2*k*x). - _Ilya Gutkovskiy_, Feb 01 2021
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 3, 5, 7, 9, 11, ...
%e 1, 11, 33, 67, 113, 171, ...
%e 1, 45, 245, 721, 1593, 2981, ...
%e 1, 195, 1921, 8179, 23649, 54691, ...
%e 1, 873, 15525, 95557, 361449, 1032801, ...
%t T[n_, k_] := Sum[If[j == k == 0, 1, k^j] * Binomial[n, j] * Binomial[2*j, j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Feb 01 2021 *)
%o (PARI) {T(n, k) = sum(j=0, n, k^j*binomial(n, j)*binomial(2*j, j))}
%o (PARI) {T(n, k) = polcoef((1+(2*k+1)*x+(k*x)^2)^n, n)}
%Y Columns k=0..3 give A000012, A026375, A084771, A340973.
%Y Rows n=0..2 give A000012, A005408, A080859.
%Y Main diagonal gives A340971.
%Y Cf. A340968.
%K nonn,tabl
%O 0,5
%A _Seiichi Manyama_, Feb 01 2021