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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).
14
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, 1, 7, 46, 245, 886, 1683, 924, 1, 1, 8, 61, 396, 1921, 5944, 8989, 3432, 1, 1, 9, 78, 595, 3606, 15525, 40636, 48639, 12870, 1, 1, 10, 97, 848, 6145, 33876, 127905, 281488, 265729, 48620, 1
OFFSET
0,5
COMMENTS
Column k is the diagonal of the rational function 1 / ((1-x)*(1-y) - k*x*y). - Seiichi Manyama, Jul 11 2020
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
LINKS
FORMULA
T(n,k) is the coefficient of x^n in the expansion of (1 + (k+1)*x + k*x^2)^n.
T(n,k) = Sum_{j=0..n} k^j * binomial(n,j)^2.
T(n,k) = Sum_{j=0..n} (k-1)^(n-j) * binomial(n,j) * binomial(n+j,j).
n * T(n,k) = (k+1) * (2*n-1) * T(n-1,k) - (k-1)^2 * (n-1) * T(n-2,k).
T(n,k) = hypergeom([-k, -k], [1], n - k), (triangular form). - Detlef Meya, May 13 2024
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, 7, ...
1, 6, 13, 22, 33, 46, 61, ...
1, 20, 63, 136, 245, 396, 595, ...
1, 70, 321, 886, 1921, 3606, 6145, ...
1, 252, 1683, 5944, 15525, 33876, 65527, ...
1, 924, 8989, 40636, 127905, 324556, 712909, ...
Seen as a triangle T(n, k):
[0] 1;
[1] 1, 1;
[2] 1, 2, 1;
[3] 1, 3, 6, 1;
[4] 1, 4, 13, 20, 1;
[5] 1, 5, 22, 63, 70, 1;
[6] 1, 6, 33, 136, 321, 252, 1;
[7] 1, 7, 46, 245, 886, 1683, 924, 1;
[8] 1, 8, 61, 396, 1921, 5944, 8989, 3432, 1;
[9] 1, 9, 78, 595, 3606, 15525, 40636, 48639, 12870, 1;
MAPLE
# Seen as a triangle read by rows:
T := (n, k) -> simplify(hypergeom([-k, -k], [1], n - k)):
seq(lprint(seq(T(n, k), k = 0..n)), n = 0..9); # Peter Luschny, May 13 2024
MATHEMATICA
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 *)
(* Seen as a triangle read by rows: *)
T[n_, k_] := HypergeometricPFQ[{-k, -k}, {1}, n - k];
Flatten[Table[T[n, k], {n, 0, 10}, {k, 0, n}]] (* Detlef Meya, May 13 2024 *)
CROSSREFS
Columns k=0..6 give A000012, A000984, A001850, A069835, A084771, A084772, A098659.
Main diagonal gives A187021.
T(n,n+1) gives A335309.
Sequence in context: A365623 A336707 A128325 * A111528 A363007 A144303
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, May 02 2019
STATUS
approved