OFFSET
0,5
LINKS
Peter Luschny, The Bell transform.
FORMULA
T(n, k) = BellMatrix([x^n] hypergeom2F0([1, 1/3], [], 3*x) / x).
T(n, k) = A371076(n, k) / k!.
From Werner Schulte, Mar 13 2024: (Start)
T(n, k) = (Sum_{i=0..k} (-1)^(k-i) * binomial(k, i) * Product_{j=0..n-1} (3*j + i)) / (k!).
T(n, k) = T(n-1, k-1) + (3*(n - 1) + k) * T(n-1, k) for 0 < k < n with initial values T(n, 0) = 0 for n > 0 and T(n, n) = 1 for n >= 0. (End)
EXAMPLE
Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 4, 1;
[3] 0, 28, 12, 1;
[4] 0, 280, 160, 24, 1;
[5] 0, 3640, 2520, 520, 40, 1;
[6] 0, 58240, 46480, 11880, 1280, 60, 1;
[7] 0, 1106560, 987840, 295960, 40040, 2660, 84, 1;
MAPLE
a := n -> mul(select(k -> k mod 3 = 1, [seq(1..3*(n + 1))])): BellMatrix(a, 9);
# Alternative:
BellMatrix(n -> coeff(series((1/x)*hypergeom([1, 1/3], [], 3*x), x, 22), x, n), 9);
# Recurrence:
T := proc(n, k) option remember; if k = n then 1 elif k = 0 then 0 else
T(n - 1, k - 1) + (3*(n - 1) + k) * T(n - 1, k) fi end:
for n from 0 to 7 do seq(T(n, k), k = 0..n) od; # Peter Luschny, Mar 13 2024
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Mar 12 2024
STATUS
approved