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Product of n-th Bell number and n-th Bell number written backwards.
1

%I #29 Sep 08 2022 08:46:21

%S 1,1,4,25,765,1300,61306,682306,1713960,1567246464,67208788225,

%T 51487177320,33511259427028,2030336608089664,42761083701194302,

%U 7549007599307190895,776831192562116876947,3388911887796350381712,649070202541887765091474,43774861324581222789850945

%N Product of n-th Bell number and n-th Bell number written backwards.

%C Conjecture: in this sequence only two semiprimes (4,25).

%F a(n) = A000110(n)*A004098(n).

%e a(4) = 765 because Bell(4) = 15 and 15*51 = 765.

%e s(5) = 1300 because Bell(5) = 52 and 52*25 = 1300.

%p b:= proc(n) option remember; `if`(n=0, 1,

%p add(b(n-j)*binomial(n-1, j-1), j=1..n))

%p end:

%p a:= n-> b(n)*(s-> parse(cat(seq(s[-i], i=1..length(s)))))(""||(b(n))):

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Apr 26 2018

%t BellB[#] FromDigits[Reverse[IntegerDigits[BellB[#]]]]&/@Range[0, 50]

%t # IntegerReverse[#]&/@BellB[Range[0,20]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Dec 29 2019 *)

%o (Magma) [Bell(n)*Seqint(Reverse(Intseq(Bell(n)))): n in [0..30]];

%o (Perl) use ntheory ":all"; sub Bell {vecsum(map{stirling($_[0],$_,2)} 0..$_[0])} for (0..30) { my $b=Bell($_); print "$_ ",vecprod($b,scalar(reverse($b))),"\n" } # _Dana Jacobsen_, Mar 04 2019

%Y Cf. A000110, A004098, A133019, A133022.

%K nonn,base,easy

%O 0,3

%A _Vincenzo Librandi_, Apr 01 2018