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A332248 Number of set partitions of [n] where all prime-indexed blocks are not singletons.

%I A332248 #19 May 08 2020 16:33:09
%S A332248 1,1,1,2,5,15,60,286,1423,7185,37758,212596,1293577,8415869,57715274,
%T A332248 414520958,3125102795,24880061105,209909409566,1871945790360,
%U A332248 17503956383037,169851122851049,1694189515772750,17248694322541778,178473482993477591,1873036127628583885
%N A332248 Number of set partitions of [n] where all prime-indexed blocks are not singletons.
%H A332248 Alois P. Heinz, <a href="/A332248/b332248.txt">Table of n, a(n) for n = 0..576</a>
%H A332248 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A332248 a(1) = 1: 1.
%e A332248 a(2) = 1: 12.
%e A332248 a(3) = 2: 123, 1|23.
%e A332248 a(4) = 5: 1234, 12|34, 13|24, 14|23, 1|234.
%e A332248 a(5) = 15: 12345, 123|45, 124|35, 125|34, 12|345, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345, 1|23|45, 1|24|35, 1|25|34.
%p A332248 b:= proc(n, i) option remember; `if`(n=0, 1, add(b(n-j, i+1)*
%p A332248        binomial(n-1, j-1), j=`if`(isprime(i), 2, 1)..n))
%p A332248     end:
%p A332248 a:= n-> b(n, 1):
%p A332248 seq(a(n), n=0..32);
%t A332248 b[n_, i_] := b[n, i] = If[n==0, 1, Sum[b[n-j, i+1] Binomial[n-1, j-1], {j, If[PrimeQ[i], 2, 1], n}]];
%t A332248 a[n_] := b[n, 1];
%t A332248 a /@ Range[0, 32] (* _Jean-François Alcover_, May 08 2020, after Maple *)
%Y A332248 Cf. A000040, A000110, A000296, A332398.
%K A332248 nonn
%O A332248 0,4
%A A332248 _Alois P. Heinz_, Feb 12 2020