A378292 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(2*n+k,r) * binomial(r,n-r)/(2*n+k) for k > 0.

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 11, 0, 1, 4, 12, 28, 46, 0, 1, 5, 18, 52, 123, 207, 0, 1, 6, 25, 84, 240, 572, 979, 0, 1, 7, 33, 125, 407, 1155, 2769, 4797, 0, 1, 8, 42, 176, 635, 2028, 5733, 13806, 24138, 0, 1, 9, 52, 238, 936, 3276, 10332, 29136, 70414, 123998, 0

Offset

0,8

Links

Table of n, a(n) for n=0..65.

Formula

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(2/k) * (1 + x * A_k(x)^(2/k)) )^k for k > 0.

G.f. of column k: B(x)^k where B(x) is the g.f. of A006605.

B(x)^k = B(x)^(k-1) + x * B(x)^(k+1) + x^2 * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k+1) + T(n-2,k+3) for n > 1.

Example

Square array begins:
  1,   1,    1,    1,     1,     1,     1, ...
  0,   1,    2,    3,     4,     5,     6, ...
  0,   3,    7,   12,    18,    25,    33, ...
  0,  11,   28,   52,    84,   125,   176, ...
  0,  46,  123,  240,   407,   635,   936, ...
  0, 207,  572, 1155,  2028,  3276,  4998, ...
  0, 979, 2769, 5733, 10332, 17140, 26860, ...

Prog

(PARI) T(n, k, t=2, u=2) = if(k==0, 0^n, k*sum(r=0, n, binomial(t*r+u*(n-r)+k, r)*binomial(r, n-r)/(t*r+u*(n-r)+k)));
matrix(7, 7, n, k, T(n-1, k-1))

Crossrefs

Columns k=0..2 give A000007, A006605, A143927.

Cf. A009766, A026300, A378289, A378290, A378291.

Sequence in context: A297328 A378289 A362079 * A055137 A143325 A307910

Adjacent sequences: A378289 A378290 A378291 * A378293 A378294 A378295

Keyword

nonn,tabl,new

Author

Seiichi Manyama, Nov 21 2024

Status

approved