A383843 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/Product_{j=0..k} (1 - j*x)^2.

1, 1, 0, 1, 2, 0, 1, 6, 3, 0, 1, 12, 23, 4, 0, 1, 20, 86, 72, 5, 0, 1, 30, 230, 480, 201, 6, 0, 1, 42, 505, 2000, 2307, 522, 7, 0, 1, 56, 973, 6300, 14627, 10044, 1291, 8, 0, 1, 72, 1708, 16464, 65002, 95060, 40792, 3084, 9, 0, 1, 90, 2796, 37632, 227542, 587580, 567240, 157440, 7181, 10, 0

Offset

0,5

Links

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

Formula

A(n,k) = Sum_{j=0..n} Stirling2(j+k,k) * Stirling2(n-j+k,k).

Example

Square array begins:
  1, 1,    1,     1,      1,       1,        1, ...
  0, 2,    6,    12,     20,      30,       42, ...
  0, 3,   23,    86,    230,     505,      973, ...
  0, 4,   72,   480,   2000,    6300,    16464, ...
  0, 5,  201,  2307,  14627,   65002,   227542, ...
  0, 6,  522, 10044,  95060,  587580,  2725380, ...
  0, 7, 1291, 40792, 567240, 4817990, 29331038, ...

Prog

(PARI) a(n, k) = sum(j=0, n, stirling(j+k, k, 2)*stirling(n-j+k, k, 2));

Crossrefs

Columns k=0..4 give A000007, A000027(n+1), A045618, A383841, A383842.

Main diagonal gives A350376.

A(n,n-1) gives A383880.

Cf. A106800, A287532.

Sequence in context: A339031 A367270 A365770 * A059299 A332673 A128722

Adjacent sequences: A383840 A383841 A383842 * A383844 A383845 A383846

Keyword

nonn,tabl

Author

Seiichi Manyama, May 12 2025

Status

approved