A316219 Number of triangles of weight prime(n) in the multiorder of integer partitions of prime numbers into prime parts.

1, 1, 3, 6, 15, 31, 92, 161, 464, 2347, 3987, 18202, 50136, 81722, 214976, 903048, 3684567, 5842249, 23206424, 57341256, 89938662, 343306266, 829972421, 3084219358, 17375700038, 40920517008, 62656899579, 146415515992, 223442878751, 518427758704, 9544240589455, 21746920337606

Offset: 1

Author

Gus Wiseman, Jun 26 2018

Keywords

• nonn

Comments

A prime partition is an integer partition of a prime number into prime parts. Then a(n) is the number of sequences of prime partitions whose sums are weakly decreasing and sum to the n-th prime number.

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• Gus Wiseman, Comcategories and Multiorders
• Gus Wiseman, Illustration of the first five terms of A316219.

Crossrefs

Cf. A000040, A000041, A000607, A056768, A063834, A100118, A269134, A281113, A316220.

Programs

Mathematica

nn=20;
pen[n_]:=pen[n]=SeriesCoefficient[Product[1/(1-x^p),{p,Select[Range[n],PrimeQ]}],{x,0,n}]
Table[Sum[Times@@pen/@p,{p,Select[IntegerPartitions[Prime[n]],And@@PrimeQ/@#&]}],{n,nn}]

PARI

P(n,f)={1/prod(k=1, n, 1 - f(k)*x^prime(k) + O(x*x^prime(n)))}
seq(n)={my(p=P(n, i->1), q=P(n, i->polcoef(p, prime(i)))); vector(n, k, polcoef(q, prime(k)))} \\ Andrew Howroyd, Jan 16 2023

Extensions

Terms a(16) and beyond from Andrew Howroyd, Jan 16 2023