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A014322 Convolution of Bell numbers with themselves.

%I A014322 #35 May 25 2025 09:11:22
%S A014322 1,2,5,14,44,154,595,2518,11591,57672,308368,1762500,10716321,
%T A014322 69011130,468856113,3348695194,25064539520,196052415230,1598543907843,
%U A014322 13556379105766,119332020447219,1088376385244908,10268343703117892,100063762955374568,1005822726810785809
%N A014322 Convolution of Bell numbers with themselves.
%C A014322 Equals row sums of triangle A144155. - _Gary W. Adamson_, Sep 12 2008
%H A014322 Alois P. Heinz, <a href="/A014322/b014322.txt">Table of n, a(n) for n = 0..576</a>
%H A014322 Adam M. Goyt and Lara K. Pudwell, <a href="http://arxiv.org/abs/1203.3786">Avoiding colored partitions of two elements in the pattern sense</a>, arXiv preprint arXiv:1203.3786 [math.CO], 2012. - From _N. J. A. Sloane_, Sep 17 2012
%F A014322 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 - ...))))))^2, a continued fraction. - _Ilya Gutkovskiy_, Sep 25 2017
%F A014322 G.f.: ( Sum_{j>=0} A000110(j)*x^j )^2. - _G. C. Greubel_, Jan 08 2023
%p A014322 with(combinat):
%p A014322 a:= n-> add(bell(i)*bell(n-i), i=0..n):
%p A014322 seq(a(n), n=0..30);  # _Alois P. Heinz_, May 13 2014
%t A014322 a[n_]:= Sum[BellB[k]*BellB[n-k], {k,0,n}];
%t A014322 Table[a[n], {n,0,30}] (* _Jean-François Alcover_, Jan 17 2016 *)
%o A014322 (Magma)
%o A014322 A014322:= func< n | (&+[Bell(j)*Bell(n-j): j in [0..n]]) >;
%o A014322 [A014322(n): n in [0..40]]; // _G. C. Greubel_, Jan 08 2023
%o A014322 (SageMath)
%o A014322 def A014322(n): return sum(bell_number(j)*bell_number(n-j) for j in range(n+1))
%o A014322 [A014322(n) for n in range(41)] # _G. C. Greubel_, Jan 08 2023
%Y A014322 Cf. A000110, A014323, A014325, A144155.
%Y A014322 Column k=2 of A292870.
%K A014322 nonn
%O A014322 0,2
%A A014322 _N. J. A. Sloane_