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A307855 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*x + (1-4*k)*x^2).

%I A307855 #39 May 13 2021 02:35:50
%S A307855 1,1,1,1,1,1,1,1,3,1,1,1,5,7,1,1,1,7,13,19,1,1,1,9,19,49,51,1,1,1,11,
%T A307855 25,91,161,141,1,1,1,13,31,145,331,581,393,1,1,1,15,37,211,561,1441,
%U A307855 2045,1107,1,1,1,17,43,289,851,2841,5797,7393,3139,1
%N A307855 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*x + (1-4*k)*x^2).
%H A307855 Seiichi Manyama, <a href="/A307855/b307855.txt">Antidiagonals n = 0..139, flattened</a>
%H A307855 T. D. Noe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html">On the Divisibility of Generalized Central Trinomial Coefficients</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
%F A307855 A(n,k) is the coefficient of x^n in the expansion of (1 + x + k*x^2)^n.
%F A307855 A(n,k) = Sum_{j=0..floor(n/2)} k^j * binomial(n,j) * binomial(n-j,j) = Sum_{j=0..floor(n/2)} k^j * binomial(n,2*j) * binomial(2*j,j).
%F A307855 D-finite with recurrence: n * A(n,k) = (2*n-1) * A(n-1,k) - (1-4*k) * (n-1) * A(n-2,k).
%e A307855 Square array begins:
%e A307855    1,   1,    1,    1,     1,     1,     1, ...
%e A307855    1,   1,    1,    1,     1,     1,     1, ...
%e A307855    1,   3,    5,    7,     9,    11,    13, ...
%e A307855    1,   7,   13,   19,    25,    31,    37, ...
%e A307855    1,  19,   49,   91,   145,   211,   289, ...
%e A307855    1,  51,  161,  331,   561,   851,  1201, ...
%e A307855    1, 141,  581, 1441,  2841,  4901,  7741, ...
%e A307855    1, 393, 2045, 5797, 12489, 22961, 38053, ...
%t A307855 T[n_, k_] := Sum[If[k == j == 0, 1, k^j] * Binomial[n, j] * Binomial[n-j, j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, May 13 2021 *)
%Y A307855 Columns k=0..6 give A000012, A002426, A084601, A084603, A084605, A098264, A098265.
%Y A307855 Main diagonal gives A187018.
%Y A307855 Cf. A110180, A292627, A307847, A307860, A307910.
%K A307855 nonn,tabl
%O A307855 0,9
%A A307855 _Seiichi Manyama_, May 01 2019