A294212 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f.: exp(Product_{j=1..n} 1/(1-x^j) - 1).

1, 1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 1, 5, 13, 0, 1, 1, 5, 25, 73, 0, 1, 1, 5, 31, 193, 501, 0, 1, 1, 5, 31, 241, 1601, 4051, 0, 1, 1, 5, 31, 265, 2261, 16741, 37633, 0, 1, 1, 5, 31, 265, 2501, 25501, 190345, 394353, 0, 1, 1, 5, 31, 265, 2621, 29461, 319915, 2509025

Offset

0,9

Links

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

Formula

B(j,k) is the coefficient of Product_{i=1..k} 1/(1-x^i).

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} j*B(j,k)*A(n-j,k)/(n-j)! for n > 0.

Example

Square array B(j,k) begins:
   1,   1,    1,    1,    1, ...
   0,   1,    1,    1,    1, ...
   0,   1,    2,    2,    2, ...
   0,   1,    2,    3,    3, ...
   0,   1,    3,    4,    5, ...
   0,   1,    3,    5,    6, ...
Square array A(n,k) begins:
   1,   1,    1,    1,    1, ...
   0,   1,    1,    1,    1, ...
   0,   3,    5,    5,    5, ...
   0,  13,   25,   31,   31, ...
   0,  73,  193,  241,  265, ...
   0, 501, 1601, 2261, 2501, ...

Crossrefs

Columns k=0..5 give A000007, A000262, A294213, A294214, A294215, A294216.

Rows n=0 gives A000012.

Main diagonal gives A058892.

Cf. A058398, A294250, A294254, A294289.

Sequence in context: A140166 A242782 A011256 * A220691 A271023 A370041

Adjacent sequences: A294209 A294210 A294211 * A294213 A294214 A294215

Keyword

nonn,tabl

Author

Seiichi Manyama, Oct 25 2017

Status

approved