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A386408 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) = Sum_{j=0..n} k^j * binomial(n,j) * Catalan(j+1).

%I A386408 #22 Aug 20 2025 10:56:31
%S A386408 1,1,1,1,3,1,1,5,10,1,1,7,29,36,1,1,9,58,185,137,1,1,11,97,532,1257,
%T A386408 543,1,1,13,146,1161,5209,8925,2219,1,1,15,205,2156,14849,53347,65445,
%U A386408 9285,1,1,17,274,3601,34041,198729,564499,491825,39587,1
%N A386408 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) = Sum_{j=0..n} k^j * binomial(n,j) * Catalan(j+1).
%F A386408 G.f. of column k: (1/x) * Series_Reversion( x/(1+(2*k+1)*x+(k*x)^2) ).
%F A386408 G.f. of column k: 2/(1 - (2*k+1)*x + sqrt((1-x) * (1-(4*k+1)*x))).
%F A386408 A(n,k) = (A340968(n+1,k) - A340968(n,k))/k for k > 0.
%F A386408 (n+2)*A(n,k) = (2*k+1)*(2*n+1)*A(n-1,k) - (4*k+1)*(n-1)*A(n-2,k) for n > 1.
%F A386408 A(n,k) = Sum_{j=0..floor(n/2)} k^(2*j) * (2*k+1)^(n-2*j) * binomial(n,2*j) * Catalan(j).
%e A386408 Square array begins:
%e A386408   1,    1,     1,      1,       1,       1,        1, ...
%e A386408   1,    3,     5,      7,       9,      11,       13, ...
%e A386408   1,   10,    29,     58,      97,     146,      205, ...
%e A386408   1,   36,   185,    532,    1161,    2156,     3601, ...
%e A386408   1,  137,  1257,   5209,   14849,   34041,    67657, ...
%e A386408   1,  543,  8925,  53347,  198729,  562551,  1330693, ...
%e A386408   1, 2219, 65445, 564499, 2748641, 9608811, 27053749, ...
%o A386408 (PARI) a(n, k) = sum(j=0, n, k^j*binomial(n, j)*(2*(j+1))!/((j+1)!*(j+2)!));
%Y A386408 Columns k=0..4 give A000012, A002212(n+1), A127846(n+1), A386362, A386389.
%Y A386408 Main diagonal gives A386432.
%Y A386408 Cf. A000108, A340968.
%K A386408 nonn,tabl
%O A386408 0,5
%A A386408 _Seiichi Manyama_, Aug 20 2025