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A307883 Square array read by descending antidiagonals: T(n, k) where column k is the expansion of 1/sqrt(1 - 2*(k+1)*x + ((k-1)*x)^2).

%I A307883 #56 May 13 2024 19:11:07
%S A307883 1,1,1,1,2,1,1,3,6,1,1,4,13,20,1,1,5,22,63,70,1,1,6,33,136,321,252,1,
%T A307883 1,7,46,245,886,1683,924,1,1,8,61,396,1921,5944,8989,3432,1,1,9,78,
%U A307883 595,3606,15525,40636,48639,12870,1,1,10,97,848,6145,33876,127905,281488,265729,48620,1
%N A307883 Square array read by descending antidiagonals: T(n, k) where column k is the expansion of 1/sqrt(1 - 2*(k+1)*x + ((k-1)*x)^2).
%C A307883 Column k is the diagonal of the rational function 1 / ((1-x)*(1-y) - k*x*y). - _Seiichi Manyama_, Jul 11 2020
%C A307883 More generally, column k is the diagonal of the rational function r / ((1-r*x)*(1-r*y) + r-1 - (k+r-1)*r*x*y) for any nonzero real number r. - _Seiichi Manyama_, Jul 22 2020
%H A307883 Seiichi Manyama, <a href="/A307883/b307883.txt">Antidiagonals n = 0..139, flattened</a>
%F A307883 T(n,k) is the coefficient of x^n in the expansion of (1 + (k+1)*x + k*x^2)^n.
%F A307883 T(n,k) = Sum_{j=0..n} k^j * binomial(n,j)^2.
%F A307883 T(n,k) = Sum_{j=0..n} (k-1)^(n-j) * binomial(n,j) * binomial(n+j,j).
%F A307883 n * T(n,k) = (k+1) * (2*n-1) * T(n-1,k) - (k-1)^2 * (n-1) * T(n-2,k).
%F A307883 T(n,k) = hypergeom([-k, -k], [1], n - k), (triangular form). - _Detlef Meya_, May 13 2024
%e A307883 Square array begins:
%e A307883   1,   1,    1,     1,      1,      1,      1, ...
%e A307883   1,   2,    3,     4,      5,      6,      7, ...
%e A307883   1,   6,   13,    22,     33,     46,     61, ...
%e A307883   1,  20,   63,   136,    245,    396,    595, ...
%e A307883   1,  70,  321,   886,   1921,   3606,   6145, ...
%e A307883   1, 252, 1683,  5944,  15525,  33876,  65527, ...
%e A307883   1, 924, 8989, 40636, 127905, 324556, 712909, ...
%e A307883 Seen as a triangle T(n, k):
%e A307883   [0] 1;
%e A307883   [1] 1, 1;
%e A307883   [2] 1, 2,  1;
%e A307883   [3] 1, 3,  6,   1;
%e A307883   [4] 1, 4, 13,  20,    1;
%e A307883   [5] 1, 5, 22,  63,   70,     1;
%e A307883   [6] 1, 6, 33, 136,  321,   252,     1;
%e A307883   [7] 1, 7, 46, 245,  886,  1683,   924,     1;
%e A307883   [8] 1, 8, 61, 396, 1921,  5944,  8989,  3432,     1;
%e A307883   [9] 1, 9, 78, 595, 3606, 15525, 40636, 48639, 12870, 1;
%p A307883 # Seen as a triangle read by rows:
%p A307883 T := (n, k) -> simplify(hypergeom([-k, -k], [1], n - k)):
%p A307883 seq(lprint(seq(T(n, k), k = 0..n)), n = 0..9);  # _Peter Luschny_, May 13 2024
%t A307883 T[n_, k_] := Sum[If[k == j == 0, 1, k^j] * Binomial[n, j]^2, {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, May 13 2021 *)
%t A307883 (* Seen as a triangle read by rows: *)
%t A307883 T[n_, k_] := HypergeometricPFQ[{-k, -k}, {1}, n - k];
%t A307883 Flatten[Table[T[n, k], {n, 0, 10}, {k, 0, n}]] (* _Detlef Meya_, May 13 2024 *)
%Y A307883 Columns k=0..6 give A000012, A000984, A001850, A069835, A084771, A084772, A098659.
%Y A307883 Main diagonal gives A187021.
%Y A307883 T(n,n+1) gives A335309.
%Y A307883 Cf. A307855, A307884, A307910.
%K A307883 nonn,tabl
%O A307883 0,5
%A A307883 _Seiichi Manyama_, May 02 2019