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A014329 Convolution of partition numbers and Catalan numbers.

%I A014329 #15 Jan 09 2023 16:41:31
%S A014329 1,2,5,12,31,84,245,752,2413,7991,27104,93605,327886,1161735,4155323,
%T A014329 14982399,54393829,198666117,729443563,2690890444,9968312790,
%U A014329 37066929338,138304185107,517646986719,1942966098461,7311862919106,27582428518833,104279585166245
%N A014329 Convolution of partition numbers and Catalan numbers.
%H A014329 Alois P. Heinz, <a href="/A014329/b014329.txt">Table of n, a(n) for n = 0..1000</a>
%F A014329 a(n) ~ c * 4^n / (sqrt(Pi) * n^(3/2)), where c = Sum_{k>=0} A000041(k)/4^k = 1/QPochhammer[1/4, 1/4] = 1.4523536424495970158347130224852748733612279788... . - _Vaclav Kotesovec_, Jun 23 2015
%F A014329 G.f.: (1 - sqrt(1-4*x))/(2*x*QPochhammer(x)). - _G. C. Greubel_, Jan 08 2023
%t A014329 Table[Sum[PartitionsP[k]*CatalanNumber[n-k],{k,0,n}],{n,0,50}] (* _Vaclav Kotesovec_, Jun 23 2015 *)
%o A014329 (Magma)
%o A014329 A000041:= func< n | NumberOfPartitions(n) >;
%o A014329 A014329:= func< n | (&+[A000041(j)*Catalan(n-j): j in [0..n]]) >;
%o A014329 [A014329(n): n in [0..40]]; // _G. C. Greubel_, Jan 08 2023
%o A014329 (SageMath)
%o A014329 def A000041(n): return number_of_partitions(n)
%o A014329 def A014329(n): return sum(A000041(j)*catalan_number(n-j) for j in range(n+1))
%o A014329 [A014329(n) for n in range(41)] # _G. C. Greubel_, Jan 08 2023
%Y A014329 Cf. A000041, A000108, A292617.
%K A014329 nonn
%O A014329 0,2
%A A014329 _N. J. A. Sloane_