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

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000041, A000108, A292617.

Programs

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

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