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A332398
Number of set partitions of [n] where all prime-indexed blocks are singletons.
2
1, 1, 2, 4, 8, 17, 40, 105, 304, 958, 3255, 11851, 46096, 191648, 854551, 4101826, 21213282, 117747119, 695773801, 4332490151, 28149712546, 189300600481, 1309755334070, 9286984108299, 67327505784439, 498502290046850, 3769028024302567, 29115361551715499
OFFSET
0,3
LINKS
EXAMPLE
a(2) = 2: 12, 1|2.
a(3) = 4: 123, 12|3, 13|2, 1|2|3.
a(4) = 8: 1234, 123|4, 124|3, 12|3|4, 134|2, 13|2|4, 14|2|3, 1|2|3|4.
a(5) = 17: 12345, 1234|5, 1235|4, 123|4|5, 1245|3, 124|3|5, 125|3|4, 12|3|4|5, 1345|2, 134|2|5, 135|2|4, 13|2|4|5, 145|2|3, 14|2|3|5, 15|2|3|4, 1|2|3|45, 1|2|3|4|5.
MAPLE
b:= proc(n, m) option remember; `if`(n=0, 1, add(`if`(j<=m
and isprime(j), 0, b(n-1, max(j, m))), j=1..m+1))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..32);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, add(b(n-j, i+1)*
binomial(n-1, j-1), j=1..`if`(isprime(i+1), 1, n)))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..32);
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n-j, i+1] Binomial[n-1, j-1], {j, 1, If[PrimeQ[i+1], 1, n]}]];
a[n_] := b[n, 0];
a /@ Range[0, 32] (* Jean-François Alcover, May 07 2020, after 2nd Maple program *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Feb 10 2020
STATUS
approved