A332248 - OEIS

%I #19 May 08 2020 16:33:09

%S 1,1,1,2,5,15,60,286,1423,7185,37758,212596,1293577,8415869,57715274,

%T 414520958,3125102795,24880061105,209909409566,1871945790360,

%U 17503956383037,169851122851049,1694189515772750,17248694322541778,178473482993477591,1873036127628583885

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

%H Alois P. Heinz, <a href="/A332248/b332248.txt">Table of n, a(n) for n = 0..576</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%e a(1) = 1: 1.

%e a(2) = 1: 12.

%e a(3) = 2: 123, 1|23.

%e a(4) = 5: 1234, 12|34, 13|24, 14|23, 1|234.

%e 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 b:= proc(n, i) option remember; `if`(n=0, 1, add(b(n-j, i+1)*

%p        binomial(n-1, j-1), j=`if`(isprime(i), 2, 1)..n))

%p     end:

%p a:= n-> b(n, 1):

%p seq(a(n), n=0..32);

%t 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 a[n_] := b[n, 1];

%t a /@ Range[0, 32] (* _Jean-François Alcover_, May 08 2020, after Maple *)

%Y Cf. A000040, A000110, A000296, A332398.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Feb 12 2020