A333988 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of (1-(k+1)*x) / (1-2*(k+1)*x+((k-1)*x)^2).

1, 1, 1, 1, 2, 1, 1, 3, 8, 1, 1, 4, 17, 32, 1, 1, 5, 28, 99, 128, 1, 1, 6, 41, 208, 577, 512, 1, 1, 7, 56, 365, 1552, 3363, 2048, 1, 1, 8, 73, 576, 3281, 11584, 19601, 8192, 1, 1, 9, 92, 847, 6016, 29525, 86464, 114243, 32768, 1, 1, 10, 113, 1184, 10033, 62976, 265721, 645376, 665857, 131072, 1

Offset

0,5

Links

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Formula

T(n,k) = Sum_{j=0..n} k^j * binomial(2*n,2*j).

T(0,k) = 1, T(1,k) = k+1 and T(n,k) = 2 * (k+1) * T(n-1,k) - (k-1)^2 * T(n-2,k) for n>1.

Example

Square array begins:
  1,   1,    1,     1,     1,     1, ...
  1,   2,    3,     4,     5,     6, ...
  1,   8,   17,    28,    41,    56, ...
  1,  32,   99,   208,   365,   576, ...
  1, 128,  577,  1552,  3281,  6016, ...
  1, 512, 3363, 11584, 29525, 62976, ...

Mathematica

T[n_, 0] := 1; T[n_, k_] := Sum[k^j * Binomial[2*n, 2*j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Sep 04 2020 *)

Prog

(PARI) {T(n, k) = sum(j=0, n, k^j*binomial(2*n, 2*j))}

Crossrefs

Column k=0..9 give A000012, A081294, A001541, A090965, A083884, A099140, A099141, A099142, A165224, A026244.

Main diagonal gives A333990.

Cf. A009999, A307883, A337389, A333989.

Sequence in context: A051467 A243716 A356776 * A195805 A293908 A346249

Adjacent sequences: A333985 A333986 A333987 * A333989 A333990 A333991

Keyword

nonn,tabl

Author

Seiichi Manyama, Sep 04 2020

Status

approved