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

0,2

Links

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

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

Mathematica

Table[Sum[PartitionsP[k]*CatalanNumber[n-k], {k, 0, n}], {n, 0, 50}] (* Vaclav Kotesovec, Jun 23 2015 *)

Prog

(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
(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

Crossrefs

Cf. A000041, A000108, A292617.

Sequence in context: A271929 A071359 A160999 * A045633 A090826 A132441

Adjacent sequences: A014326 A014327 A014328 * A014330 A014331 A014332

Keyword

nonn

Author

N. J. A. Sloane

Status

approved