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%I #23 Jul 21 2023 14:05:08
%S 1,2,1,1,36,1,216,900,1,1296,5400,44100,27000,7776,32400,264600,
%T 5336100,162000,1323000,46656,194400,810000,1587600,9261000,32016600,
%U 901800900,972000,7938000,160083000,279936,1166400,4860000,9525600,39690000,55566000,192099600,1120581000
%N Prime encoded sequence of generic integer partitions of n in the antilexicographic order of the partitions.
%C By definition [1] is a generic partition and 0 has no generic partitions. For n > 1 a partition p of n is generic if it does not have the form [1+r_1,r_2,...,r_k] or [r_1,r_2,...,r_k,1] for some partition [r_1,r_2,...,r_k] of n-1.
%C Encoding: The partition p = [p_1,...,p_k] is represented by Product_{i=1..k} prime(i) ^ p_i. If n has generic partitions then these encodings are listed in the antilexicographic order of the partitions; if n has no generic partitions then this fact is represented by '1'.
%C Starting from generic partitions a table of all partitions can be built by two operations: appending '1' at the tail of a partition or adding 1 to the head of a partition (see the table at the link given).
%C A generic partition is a partition of the form [x,x,p_2,...,p_k-1,y] with y > 1; in addition [1] is a generic partition by definition.
%H Alois P. Heinz, <a href="/A182911/b182911.txt">Table of n, a(n) for n = 0..19200</a>
%H Peter Luschny, Integer partition trees, <a href="http://oeis.org/wiki/User:Peter_Luschny/IntegerPartitionTrees">OEIS wiki</a>.
%e 0: {} -> 1
%e 1: {[1]} -> 2^1 = 2
%e 2: {} -> 1
%e 3: {} -> 1
%e 4: {[22]} -> 2^2*3^2 = 36
%e 5: {} -> 1
%e 6: {[33],[222]} -> 2^3*3^3 = 216; 2^2*3^2*5^2 = 900
%e 7: {} -> 1
%e 8: {[44],[332],[2222]} -> 1296, 5400, 44100
%e 9: {[333]} -> 27000
%p a:= proc(n) local b, ll; b:=
%p proc(n,i,l) local nl; nl:= nops(l);
%p if n<0 then
%p elif n=0 then ll:= ll,
%p `if`(nl=0 or nl=1 and l[1]=1 or
%p nl>1 and l[-1]<>1 and l[1]=l[2],
%p mul(ithprime(t)^l[t], t=1..nl), NULL)
%p elif i=0 then
%p else b(n-i, i, [l[], i]), b(n, i-1, l)
%p fi
%p end;
%p ll:= NULL; b(n,n,[]);
%p `if`(ll=NULL,1,ll)
%p end:
%p seq(a(n), n=0..15);
%Y Cf. A046056, A053445.
%K nonn
%O 0,2
%A _Peter Luschny_, Jan 26 2011
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