OFFSET
0,8
COMMENTS
A(n,k) is the n-th term of the k-fold convolution of Bell numbers with themselves. - Alois P. Heinz, Feb 12 2019
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
FORMULA
G.f. of column k: (1/(1 - x - x^2/(1 - 2*x - 2*x^2/(1 - 3*x - 3*x^2/(1 - 4*x - 4*x^2/(1 - ...))))))^k, a continued fraction.
EXAMPLE
G.f. of column k: A_k(x) = 1 + k*x + k*(k + 3)*x^2/2 + k*(k^2 + 9*k + 20)*x^3/6 + k*(k^3 + 18*k^2 + 107*k + 234)*x^4/24 + k*(k^4 + 30*k^3 + 335*k^2 + 1770*k + 4104)*x^5/120 + ...
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 2, 5, 9, 14, 20, ...
0, 5, 14, 28, 48, 75, ...
0, 15, 44, 93, 169, 280, ...
0, 52, 154, 333, 624, 1071, ...
MAPLE
A:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0, 0,
`if`(k=1, add(A(n-j, k)*binomial(n-1, j-1), j=1..n),
(h-> add(A(j, h)*A(n-j, k-h), j=0..n))(iquo(k, 2)))))
end:
seq(seq(A(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, May 31 2018
MATHEMATICA
Table[Function[k, SeriesCoefficient[1/(1 - x + ContinuedFractionK[-i x^2, 1 - (i + 1) x, {i, 1, n}])^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Ilya Gutkovskiy, Sep 25 2017
STATUS
approved