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

1,3

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.

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}]

Prog

(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

Crossrefs

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

Sequence in context: A244711 A244712 A006961 * A034740 A367293 A323936

Adjacent sequences: A316216 A316217 A316218 * A316220 A316221 A316222

Keyword

nonn

Author

Gus Wiseman, Jun 26 2018

Extensions

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

Status

approved