The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #44 Aug 27 2025 06:32:48
%S 1,1,1,1,2,1,1,3,5,1,1,4,13,15,1,1,5,25,71,51,1,1,6,41,199,441,188,1,
%T 1,7,61,429,1795,2955,731,1,1,8,85,791,5073,17422,20805,2950,1,1,9,
%U 113,1315,11571,64469,177463,151695,12235,1
%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)*Catalan(j).
%H Seiichi Manyama, <a href="/A340968/b340968.txt">Antidiagonals n = 0..139, flattened</a>
%F G.f. A_k(x) of column k satisfies A_k(x) = 1/(1 - x) + k*x*A_k(x)^2.
%F A_k(x) = 2/( 1 - x + sqrt((1 - x) * (1 - (4*k+1)*x)) ).
%F T(n,k) = 1 + k * Sum_{j=0..n-1} T(j,k) * T(n-1-j,k).
%F (n+1) * T(n,k) = 2 * ((2*k+1) * n - k) * 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) - BesselI(1,2*k*x)). - _Ilya Gutkovskiy_, Feb 01 2021
%F T_row(n) = k -> hypergeom([1/2, -n], [2], -4*k). - _Peter Luschny_, Aug 27 2025
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 2, 3, 4, 5, 6, ...
%e 1, 5, 13, 25, 41, 61, ...
%e 1, 15, 71, 199, 429, 791, ...
%e 1, 51, 441, 1795, 5073, 11571, ...
%e 1, 188, 2955, 17422, 64469, 181776, ...
%p T_row := n -> k -> hypergeom([1/2, -n], [2], -4*k): for n from 0 to 6 do seq(simplify(T_row(n)(k)), k = 0..6) od; # _Peter Luschny_, Aug 27 2025
%t T[n_, k_] := Sum[If[j == k == 0, 1, k^j] * Binomial[n, j] * CatalanNumber[j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Feb 01 2021 *)
%t A340968[n_, k_] := Hypergeometric2F1[1/2, -n, 2, -4*k]; Table[A340968[n, k], {n, 0, 6}, {k, 0, 7}] (* row-wise *) (* _Peter Luschny_, Aug 27 2025 *)
%o (PARI) T(n, k) = sum(j=0, n, k^j*binomial(n, j)*(2*j)!/(j!*(j+1)!));
%o (PARI) T(n, k) = 1+k*sum(j=0, n-1, T(j, k)*T(n-1-j, k));
%Y Columns k=0..4 give A000012, A007317(n+1), A162326(n+1), A337167, A386387.
%Y Main diagonal gives A338979.
%Y Cf. A000108, A290605, A340970.
%K nonn,tabl
%O 0,5
%A _Seiichi Manyama_, Jan 31 2021