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A343667
Number of partitions of an n-set without blocks of size 7.
7
1, 1, 2, 5, 15, 52, 203, 876, 4132, 21075, 115375, 673620, 4172413, 27296089, 187891174, 1356343385, 10238632307, 80615222404, 660560758879, 5621465069117, 49594663447612, 452846969975391, 4273130715906123, 41612346388251187, 417668648929556073, 4315893703814296053
OFFSET
0,3
FORMULA
E.g.f.: exp(exp(x) - 1 - x^7/7!).
a(n) = n! * Sum_{k=0..floor(n/7)} (-1)^k * Bell(n-7*k) / ((n-7*k)! * k! * (7!)^k).
MAPLE
a:= proc(n) option remember; `if`(n=0, 1, add(
`if`(j=7, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
end:
seq(a(n), n=0..25); # Alois P. Heinz, Apr 25 2021
MATHEMATICA
nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^7/7!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 7 k]/((n - 7 k)! k! (7!)^k), {k, 0, Floor[n/7]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 7, 0, Binomial[n - 1, k - 1] a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 25 2021
STATUS
approved