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A306684 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - x + sqrt(1 - 2*x + (1-4*k)*x^2)).

%I A306684 #36 May 12 2021 03:54:53
%S A306684 1,1,1,1,1,1,1,1,2,1,1,1,3,4,1,1,1,4,7,9,1,1,1,5,10,21,21,1,1,1,6,13,
%T A306684 37,61,51,1,1,1,7,16,57,121,191,127,1,1,1,8,19,81,201,451,603,323,1,1,
%U A306684 1,9,22,109,301,861,1639,1961,835,1
%N A306684 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - x + sqrt(1 - 2*x + (1-4*k)*x^2)).
%H A306684 Seiichi Manyama, <a href="/A306684/b306684.txt">Antidiagonals n = 0..139, flattened</a>
%F A306684 A(n,k) is the coefficient of x^n in the expansion of 1/(n+1) * (1 + x + k*x^2)^(n+1).
%F A306684 A(n,k) = Sum_{j=0..floor(n/2)} k^j * binomial(n,j) * binomial(n-j,j)/(j+1) = Sum_{j=0..floor(n/2)} k^j * binomial(n,2*j) * A000108(j).
%F A306684 (n+2) * A(n,k) = (2*n+1) * A(n-1,k) - (1-4*k) * (n-1) * A(n-2,k).
%e A306684 Square array begins:
%e A306684    1,   1,   1,    1,    1,    1,     1,     1, ...
%e A306684    1,   1,   1,    1,    1,    1,     1,     1, ...
%e A306684    1,   2,   3,    4,    5,    6,     7,     8, ...
%e A306684    1,   4,   7,   10,   13,   16,    19,    22, ...
%e A306684    1,   9,  21,   37,   57,   81,   109,   141, ...
%e A306684    1,  21,  61,  121,  201,  301,   421,   561, ...
%e A306684    1,  51, 191,  451,  861, 1451,  2251,  3291, ...
%e A306684    1, 127, 603, 1639, 3445, 6231, 10207, 15583, ...
%t A306684 T[n_, k_] := Sum[If[k == j == 0, 1, k^j] * Binomial[n, 2*j] * CatalanNumber[j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, May 12 2021 *)
%Y A306684 Columns k=0..7 give A000012, A001006, A025235, A025237, A091147, A091148, A091149, A217275.
%Y A306684 Main diagonal gives A307906.
%Y A306684 Cf. A000108, A107267, A247495, A307855.
%K A306684 nonn,tabl
%O A306684 0,9
%A A306684 _Seiichi Manyama_, May 06 2019