A014329 Convolution of partition numbers and Catalan numbers.
1, 2, 5, 12, 31, 84, 245, 752, 2413, 7991, 27104, 93605, 327886, 1161735, 4155323, 14982399, 54393829, 198666117, 729443563, 2690890444, 9968312790, 37066929338, 138304185107, 517646986719, 1942966098461, 7311862919106, 27582428518833, 104279585166245
Offset: 0
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
Programs
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Magma
A000041:= func< n | NumberOfPartitions(n) >; A014329:= func< n | (&+[A000041(j)*Catalan(n-j): j in [0..n]]) >; [A014329(n): n in [0..40]]; // G. C. Greubel, Jan 08 2023
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Mathematica
Table[Sum[PartitionsP[k]*CatalanNumber[n-k],{k,0,n}],{n,0,50}] (* Vaclav Kotesovec, Jun 23 2015 *) -
SageMath
def A000041(n): return number_of_partitions(n) def A014329(n): return sum(A000041(j)*catalan_number(n-j) for j in range(n+1)) [A014329(n) for n in range(41)] # G. C. Greubel, Jan 08 2023
Formula
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
G.f.: (1 - sqrt(1-4*x))/(2*x*QPochhammer(x)). - G. C. Greubel, Jan 08 2023