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A288790
Number of blocks of size >= eight in all set partitions of n.
2
1, 10, 101, 947, 8670, 79249, 730745, 6838642, 65197797, 634656360, 6316333291, 64318009411, 670336612614, 7151290120037, 78085166445577, 872478836270306, 9972817907218608, 116575837400037486, 1393037460835481622, 17010118386233081680, 212160149063581345610
OFFSET
8,2
LINKS
FORMULA
a(n) = Bell(n+1) - Sum_{j=0..7} binomial(n,j) * Bell(n-j).
a(n) = Sum_{j=0..n-8} binomial(n,j) * Bell(j).
E.g.f.: (exp(x) - Sum_{k=0..7} x^k/k!) * exp(exp(x) - 1). - Ilya Gutkovskiy, Jun 26 2022
MAPLE
b:= proc(n) option remember; `if`(n=0, 1, add(
b(n-j)*binomial(n-1, j-1), j=1..n))
end:
g:= proc(n, k) option remember; `if`(n<k, 0,
g(n, k+1) +binomial(n, k)*b(n-k))
end:
a:= n-> g(n, 8):
seq(a(n), n=8..30);
MATHEMATICA
Table[Sum[Binomial[n, j] BellB[j], {j, 0, n - 8}], {n, 8, 30}] (* Indranil Ghosh, Jul 06 2017 *)
PROG
(Python)
from sympy import bell, binomial
def a(n): return sum(binomial(n, j)*bell(j) for j in range(n - 7))
print([a(n) for n in range(8, 31)]) # Indranil Ghosh, Jul 06 2017
CROSSREFS
Column k=8 of A283424.
Cf. A000110.
Sequence in context: A214390 A293804 A210167 * A280371 A105032 A281102
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 15 2017
STATUS
approved