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%I #11 Sep 15 2016 09:37:26
%S 1,1,2,4,11,30,122,336,1412,15129,44561,417542,2479120,7540843,
%T 35983502,451454834,5313515136,16809858904,190077477328,1124302066470,
%U 3521811953565,38563707677633,240966297786218,3192420711942298,95433674596402663,567734580765228356
%N Number of prime plane trees of weight prime(n).
%C A prime plane tree is either (case 1) a prime number, or (case 2) a sequence of prime plane trees whose weights are an integer partition of a prime number, where the weight of a tree is the sum of weights of its branches. Prime plane trees are "multichains" in the multiorder of integer partitions of prime numbers into prime parts (A056768).
%H Alois P. Heinz, <a href="/A276687/b276687.txt">Table of n, a(n) for n = 1..681</a>
%H Gus Wiseman, <a href="https://docs.google.com/document/d/1m0s6DGTBkDW9gvMuFmJHvy6oLGRAbQ7okAZcOPZawp0/pub">Comcategories and Multiorders</a>, <a href="http://www.nafindix.com/math/academic/ComcategoriesandMultiordersv7.pdf">(pdf version)</a>
%e The a(5) = 11 prime plane trees of weight A000040(5) = 11 are: {11, (3,3,5), (3,3,(2,3)), (2,2,7), (2,2,(2,5)), (2,2,(2,(2,3))), (2,2,(2,2,3)), (2,3,3,3), (2,2,2,5), (2,2,2,(2,3)), (2,2,2,2,3)}.
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=2, 0,
%p b(n, prevprime(i)))+`if`(i>n, 0, b(n-i, i)*(1+
%p `if`(i>2, b(i, prevprime(i)), 0))))
%p end:
%p a:= n-> `if`(n<3, 1, 1+b(ithprime(n), ithprime(n-1))):
%p seq(a(n), n=1..40); # _Alois P. Heinz_, Sep 15 2016
%t n=20;
%t ser=Product[1/(1-c[Prime[i]]*x^Prime[i]),{i,1,n}];
%t sys=Table[c[Prime[i]]==Expand[SeriesCoefficient[ser,{x,0,Prime[i]}]-c[Prime[i]]+1],{i,1,n}];
%t Block[{c},Set@@@sys]
%Y Cf. A000040, A056768, A000607, A100118, A196545, A273873.
%K nonn
%O 1,3
%A _Gus Wiseman_, Sep 13 2016
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