A373717 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..floor(k*n/(2*k+1))} binomial(k * (n-2*j),j).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 4, 5, 4, 1, 1, 1, 1, 5, 7, 8, 6, 1, 1, 1, 1, 6, 9, 13, 15, 9, 1, 1, 1, 1, 7, 11, 19, 28, 26, 13, 1, 1, 1, 1, 8, 13, 26, 45, 53, 45, 19, 1, 1, 1, 1, 9, 15, 34, 66, 91, 105, 80, 28, 1, 1, 1, 1, 10, 17, 43, 91, 141, 201, 211, 140, 41, 1

Offset

0,14

Links

Table of n, a(n) for n=0..90.

Formula

G.f. of column k: 1/(1 - x * (1 + x^2)^k).

T(n,k) = Sum_{j=0..k} binomial(k,j) * T(n-2*j-1,k).

Example

Square array begins:
  1, 1,  1,  1,  1,  1,  1, ...
  1, 1,  1,  1,  1,  1,  1, ...
  1, 1,  1,  1,  1,  1,  1, ...
  1, 2,  3,  4,  5,  6,  7, ...
  1, 3,  5,  7,  9, 11, 13, ...
  1, 4,  8, 13, 19, 26, 34, ...
  1, 6, 15, 28, 45, 66, 91, ...

Prog

(PARI) T(n, k) = sum(j=0, k*n\(2*k+1), binomial(k*(n-2*j), j));

Crossrefs

Columns k=0..3 give A000012, A000930, A193147, A373718.

Main diagonal gives A373719.

Cf. A099233.

Sequence in context: A181386 A193517 A296554 * A377007 A327482 A189006

Adjacent sequences: A373714 A373715 A373716 * A373718 A373719 A373720

Keyword

nonn,tabl

Author

Seiichi Manyama, Jun 15 2024

Status

approved