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Three-fold convolution of Bell numbers with themselves.
5

%I #17 Jan 09 2023 01:41:18

%S 1,3,9,28,93,333,1289,5394,24366,118526,618924,3456942,20573391,

%T 129951231,867877107,6106194478,45109290477,348836705235,

%U 2816093142803,23673989688810,206794355179656,1873232870155036,17565534522745008,170237112831874188

%N Three-fold convolution of Bell numbers with themselves.

%H Alois P. Heinz, <a href="/A014323/b014323.txt">Table of n, a(n) for n = 0..576</a>

%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 - ...))))))^3, a continued fraction. - _Ilya Gutkovskiy_, Sep 25 2017

%F From _G. C. Greubel_, Jan 08 2023: (Start)

%F a(n) = Sum_{j=0..n} A000110(j)*A014322(n-j).

%F G.f.: ( Sum_{j>=0} A000110(j)*x^j )^3. (End)

%t A014322[n_]:= Sum[BellB[j]*BellB[n-j], {j,0,n}];

%t A014323[n_]:= Sum[BellB[j]*A014322[n-j], {j,0,n}];

%t Table[A014323[n], {n,0,40}] (* _G. C. Greubel_, Jan 08 2023 *)

%o (Magma)

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

%o A014323:= func< n | (&+[Bell(j)*A014322(n-j): j in [0..n]]) >;

%o [A014323(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 def A014323(n): return sum(bell_number(j)*A014322(n-j) for j in range(n+1))

%o [A014323(n) for n in range(41)] # _G. C. Greubel_, Jan 08 2023

%Y Cf. A000110, A014322, A014325.

%Y Column k=3 of A292870.

%K nonn

%O 0,2

%A _N. J. A. Sloane_