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Convolution of Bell and Catalan numbers.
1

%I #13 Jan 09 2023 02:23:04

%S 1,2,5,14,43,143,512,1974,8226,37224,183456,984098,5719900,35767592,

%T 238720688,1688044543,12568879291,98065500372,798734909795,

%U 6771216844711,59602783525634,543665320690323,5129940111134397,49997388546860666,502624275694700979

%N Convolution of Bell and Catalan numbers.

%H Alois P. Heinz, <a href="/A014327/b014327.txt">Table of n, a(n) for n = 0..576</a>

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

%F a(n) = Sum_{j=0..n} A000110(j)*A000108(n-j).

%F G.f.: (1/(2*x))*(1 - sqrt(1-4*x))*Sum_{j>=0} A000110(j)*x^j. (End)

%t A014327[n_]:= A014327[n]= Sum[BellB[j]*CatalanNumber[n-j], {j,0,n}];

%t Table[A014327[n], {n,0,40}] (* _G. C. Greubel_, Jan 08 2023 *)

%o (Magma)

%o A014327:= func< n | (&+[Bell(j)*Catalan(n-j): j in [0..n]]) >;

%o [A014327(n): n in [0..40]]; // _G. C. Greubel_, Jan 08 2023

%o (SageMath)

%o def A014327(n): return sum(bell_number(j)*catalan_number(n-j) for j in range(n+1))

%o [A014327(n) for n in range(41)] # _G. C. Greubel_, Jan 08 2023

%Y Cf. A000108, A000110.

%K nonn

%O 0,2

%A _N. J. A. Sloane_