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Convolution of Bell numbers with themselves.
8

%I #31 Jan 09 2023 01:43:37

%S 1,2,5,14,44,154,595,2518,11591,57672,308368,1762500,10716321,

%T 69011130,468856113,3348695194,25064539520,196052415230,1598543907843,

%U 13556379105766,119332020447219,1088376385244908,10268343703117892,100063762955374568,1005822726810785809

%N Convolution of Bell numbers with themselves.

%C Equals row sums of triangle A144155. - _Gary W. Adamson_, Sep 12 2008

%H Alois P. Heinz, <a href="/A014322/b014322.txt">Table of n, a(n) for n = 0..576</a>

%H 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 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 G.f.: ( Sum_{j>=0} A000110(j)*x^j )^2. - _G. C. Greubel_, Jan 08 2023

%p with(combinat):

%p a:= n-> add(bell(i)*bell(n-i), i=0..n):

%p seq(a(n), n=0..30); # _Alois P. Heinz_, May 13 2014

%t a[n_]:= Sum[BellB[k]*BellB[n-k], {k,0,n}];

%t Table[a[n], {n,0,30}] (* _Jean-François Alcover_, Jan 17 2016 *)

%o (Magma)

%o A014322:= func< n | (&+[Bell(j)*Bell(n-j): j in [0..n]]) >;

%o [A014322(n): n in [0..40]]; // _G. C. Greubel_, Jan 08 2023

%o (SageMath)

%o def A014322(n): return sum(bell_number(j)*bell_number(n-j) for j in range(n+1))

%o [A014322(n) for n in range(41)] # _G. C. Greubel_, Jan 08 2023

%Y Cf. A000110, A014323, A014325, A144155.

%Y Column k=2 of A292870.

%K nonn

%O 0,2

%A _N. J. A. Sloane_