OFFSET
1,4
COMMENTS
The Bell transform of A001187(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 17 2016
LINKS
Alois P. Heinz, Rows n = 1..82, flattened (first 25 rows from Marko Riedel)
Marko Riedel, Maple implementation of memoized recurrence.
Marko Riedel et al., Proof of recurrence relation.
FORMULA
SUM[n,k=0..oo] T(n,k) * x^n * y^k / n! = exp( y*( F(x) - 1 ) ) = ( SUM[n=0..oo] 2^binomial(n, 2)*x^n/n! )^y, where F(x) is e.g.f. of A001187.
T(n,k) = Sum_{q=0..n-1} C(n-1, q) T(q, k-1) 2^C(n-q,2) - Sum_{q=0..n-2} C(n-1, q) T(q+1, k) 2^C(n-1-q, 2) where T(0,0) = 1 and T(0,k) = 0 and T(n,0) = 0. - Marko Riedel, Feb 04 2019
Sum_{k=1..n} k * T(n,k) = A125207(n) - Alois P. Heinz, Feb 02 2024
EXAMPLE
The triangle T(n,k) starts as:
n=1: 1;
n=2: 1, 1;
n=3: 4, 3, 1;
n=4: 38, 19, 6, 1;
n=5: 728, 230, 55, 10, 1;
n=6: 26704, 5098, 825, 125, 15, 1;
...
MAPLE
g:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-add(
binomial(n, k)*2^((n-k)*(n-k-1)/2)*g(k)*k, k=1..n-1)/n)
end:
b:= proc(n) option remember; `if`(n=0, 1, add(expand(
b(n-j)*binomial(n-1, j-1)*g(j)*x), j=1..n))
end:
T:= (n, k)-> coeff(b(n$2), x, k):
seq(seq(T(n, k), k=1..n), n=1..10); # Alois P. Heinz, Feb 02 2024
MATHEMATICA
a= Sum[2^Binomial[n, 2] x^n/n!, {n, 0, 10}];
Rest[Transpose[Table[Range[0, 10]! CoefficientList[Series[Log[a]^n/n!, {x, 0, 10}], x], {n, 1, 10}]]] // Grid (* Geoffrey Critzer, Mar 15 2011 *)
PROG
# Adds a column 1, 0, 0, 0, ... at the left side of the triangle.
bell_matrix(lambda n: A001187(n+1), 9) # Peter Luschny, Jan 17 2016
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Max Alekseyev, Aug 23 2008
STATUS
approved