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A014323 Three-fold convolution of Bell numbers with themselves.

%I A014323 #17 Jan 09 2023 01:41:18
%S A014323 1,3,9,28,93,333,1289,5394,24366,118526,618924,3456942,20573391,
%T A014323 129951231,867877107,6106194478,45109290477,348836705235,
%U A014323 2816093142803,23673989688810,206794355179656,1873232870155036,17565534522745008,170237112831874188
%N A014323 Three-fold convolution of Bell numbers with themselves.
%H A014323 Alois P. Heinz, <a href="/A014323/b014323.txt">Table of n, a(n) for n = 0..576</a>
%F A014323 G.f.: (1/(1 - x - x^2/(1 - 2*x - 2*x^2/(1 - 3*x - 3*x^2/(1 - 4*x - 4*x^2/(1 - ...))))))^3, a continued fraction. - _Ilya Gutkovskiy_, Sep 25 2017
%F A014323 From _G. C. Greubel_, Jan 08 2023: (Start)
%F A014323 a(n) = Sum_{j=0..n} A000110(j)*A014322(n-j).
%F A014323 G.f.: ( Sum_{j>=0} A000110(j)*x^j )^3. (End)
%t A014323 A014322[n_]:= Sum[BellB[j]*BellB[n-j], {j,0,n}];
%t A014323 A014323[n_]:= Sum[BellB[j]*A014322[n-j], {j,0,n}];
%t A014323 Table[A014323[n], {n,0,40}] (* _G. C. Greubel_, Jan 08 2023 *)
%o A014323 (Magma)
%o A014323 A014322:= func< n | (&+[Bell(j)*Bell(n-j): j in [0..n]]) >;
%o A014323 A014323:= func< n | (&+[Bell(j)*A014322(n-j): j in [0..n]]) >;
%o A014323 [A014323(n): n in [0..40]]; // _G. C. Greubel_, Jan 08 2023
%o A014323 (SageMath)
%o A014323 def A014322(n): return sum(bell_number(j)*bell_number(n-j) for j in range(n+1))
%o A014323 def A014323(n): return sum(bell_number(j)*A014322(n-j) for j in range(n+1))
%o A014323 [A014323(n) for n in range(41)] # _G. C. Greubel_, Jan 08 2023
%Y A014323 Cf. A000110, A014322, A014325.
%Y A014323 Column k=3 of A292870.
%K A014323 nonn
%O A014323 0,2
%A A014323 _N. J. A. Sloane_