A385017 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A385013.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 4, 0, 1, 4, 9, 12, 9, 0, 1, 5, 14, 25, 30, 19, 0, 1, 6, 20, 44, 69, 72, 37, 0, 1, 7, 27, 70, 133, 183, 164, 52, 0, 1, 8, 35, 104, 230, 384, 464, 326, -25, 0, 1, 9, 44, 147, 369, 716, 1060, 1083, 435, -630, 0, 1, 10, 54, 200, 560, 1230, 2125, 2748, 2139, -464, -3616, 0

Offset

0,8

Links

Table of n, a(n) for n=0..77.

Formula

Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} binomial(-2*n+2*j+k-1,j-1) * b(n-j,j)/j. Then A(n,k) = b(n,-k).

Example

Square array begins:
  1,  1,   1,   1,    1,    1,    1, ...
  0,  1,   2,   3,    4,    5,    6, ...
  0,  2,   5,   9,   14,   20,   27, ...
  0,  4,  12,  25,   44,   70,  104, ...
  0,  9,  30,  69,  133,  230,  369, ...
  0, 19,  72, 183,  384,  716, 1230, ...
  0, 37, 164, 464, 1060, 2125, 3893, ...

Prog

(PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-2*n+2*j+k-1, j-1)*b(n-j, j)/j));
a(n, k) = b(n, -k);

Crossrefs

Columns k=0..1 give A000007, A385013.

Cf. A384976.

Sequence in context: A144064 A172236 A191646 * A297321 A277938 A130020

Adjacent sequences: A385014 A385015 A385016 * A385018 A385019 A385020

Keyword

sign,tabl

Author

Seiichi Manyama, Jun 15 2025

Status

approved