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Convolution of partition numbers and Catalan numbers.
6

%I #15 Jan 09 2023 16:41:31

%S 1,2,5,12,31,84,245,752,2413,7991,27104,93605,327886,1161735,4155323,

%T 14982399,54393829,198666117,729443563,2690890444,9968312790,

%U 37066929338,138304185107,517646986719,1942966098461,7311862919106,27582428518833,104279585166245

%N Convolution of partition numbers and Catalan numbers.

%H Alois P. Heinz, <a href="/A014329/b014329.txt">Table of n, a(n) for n = 0..1000</a>

%F 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 G.f.: (1 - sqrt(1-4*x))/(2*x*QPochhammer(x)). - _G. C. Greubel_, Jan 08 2023

%t Table[Sum[PartitionsP[k]*CatalanNumber[n-k],{k,0,n}],{n,0,50}] (* _Vaclav Kotesovec_, Jun 23 2015 *)

%o (Magma)

%o A000041:= func< n | NumberOfPartitions(n) >;

%o A014329:= func< n | (&+[A000041(j)*Catalan(n-j): j in [0..n]]) >;

%o [A014329(n): n in [0..40]]; // _G. C. Greubel_, Jan 08 2023

%o (SageMath)

%o def A000041(n): return number_of_partitions(n)

%o def A014329(n): return sum(A000041(j)*catalan_number(n-j) for j in range(n+1))

%o [A014329(n) for n in range(41)] # _G. C. Greubel_, Jan 08 2023

%Y Cf. A000041, A000108, A292617.

%K nonn

%O 0,2

%A _N. J. A. Sloane_