cp's OEIS Frontend

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

%I A316219 #12 Jan 16 2023 23:10:52
%S A316219 1,1,3,6,15,31,92,161,464,2347,3987,18202,50136,81722,214976,903048,
%T A316219 3684567,5842249,23206424,57341256,89938662,343306266,829972421,
%U A316219 3084219358,17375700038,40920517008,62656899579,146415515992,223442878751,518427758704,9544240589455,21746920337606
%N A316219 Number of triangles of weight prime(n) in the multiorder of integer partitions of prime numbers into prime parts.
%C A316219 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.
%H A316219 Andrew Howroyd, <a href="/A316219/b316219.txt">Table of n, a(n) for n = 1..1000</a>
%H A316219 Gus Wiseman, <a href="https://docs.google.com/document/d/1m0s6DGTBkDW9gvMuFmJHvy6oLGRAbQ7okAZcOPZawp0/pub">Comcategories and Multiorders</a>
%H A316219 Gus Wiseman, <a href="/A316219/a316219.png">Illustration of the first five terms of A316219.</a>
%t A316219 nn=20;
%t A316219 pen[n_]:=pen[n]=SeriesCoefficient[Product[1/(1-x^p),{p,Select[Range[n],PrimeQ]}],{x,0,n}]
%t A316219 Table[Sum[Times@@pen/@p,{p,Select[IntegerPartitions[Prime[n]],And@@PrimeQ/@#&]}],{n,nn}]
%o A316219 (PARI)
%o A316219 P(n,f)={1/prod(k=1, n, 1 - f(k)*x^prime(k) + O(x*x^prime(n)))}
%o A316219 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
%Y A316219 Cf. A000040, A000041, A000607, A056768, A063834, A100118, A269134, A281113, A316220.
%K A316219 nonn
%O A316219 1,3
%A A316219 _Gus Wiseman_, Jun 26 2018
%E A316219 Terms a(16) and beyond from _Andrew Howroyd_, Jan 16 2023