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 A316219 Number of triangles of weight prime(n) in the multiorder of integer partitions of prime numbers into prime parts. 5
 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 (list; graph; refs; listen; history; text; internal format)
 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 A323936 A305839 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

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Last modified November 29 06:17 EST 2023. Contains 367422 sequences. (Running on oeis4.)