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%I #18 May 19 2021 05:14:19
%S 1,1,4,10,34,80,254,604,1785,4370,11986,29286,80355,193137,505952,
%T 1239348,3181970,7686199,19520906,46931241,117334784,282021070,
%U 693721166,1659075192,4063164983,9651686516,23347635094,55405326513,133021397071,313842472333,749299686508
%N Number of thrice-partitions of n.
%C A thrice-partition of n is a choice of a twice-partition of each part in a partition of n. Thrice-partitions correspond to intervals in the lattice form of the multiorder of integer partitions.
%H Alois P. Heinz, <a href="/A301595/b301595.txt">Table of n, a(n) for n = 0..3244</a>
%H Gus Wiseman, <a href="/A301595/a301595.png">The a(4) = 34 thrice-partitions of 4.</a>
%F O.g.f.: Product_{n > 0} 1/(1 - A063834(n) x^n).
%e The a(3) = 10 thrice-partitions:
%e ((3)), ((21)), ((111)), ((2)(1)), ((11)(1)), ((1)(1)(1)),
%e ((2))((1)), ((11))((1)), ((1)(1))((1)),
%e ((1))((1))((1)).
%p b:= proc(n, i, k) option remember; `if`(n=0 or k=0 or i=1,
%p 1, b(n, i-1, k)+b(i$2, k-1)*b(n-i, min(n-i, i), k))
%p end:
%p a:= n-> b(n$2, 3):
%p seq(a(n), n=0..35); # _Alois P. Heinz_, Jan 25 2019
%t twie[n_]:=Sum[Times@@PartitionsP/@ptn,{ptn,IntegerPartitions[n]}];
%t thrie[n_]:=Sum[Times@@twie/@ptn,{ptn,IntegerPartitions[n]}];
%t Array[thrie,30]
%t (* Second program: *)
%t b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0 || i == 1,
%t 1, b[n, i - 1, k] + b[i, i, k - 1]*b[n - i, Min[n - i, i], k]];
%t a[n_] := b[n, n, 3];
%t a /@ Range[0, 35] (* _Jean-François Alcover_, May 19 2021, after _Alois P. Heinz_ *)
%Y Cf. A000041, A001383, A001970, A061260, A063834, A119442, A196545, A281113, A289501, A300383, A301422, A301462, A301480, A301595, A301598, A301706.
%Y Column k=3 of A323718.
%K nonn
%O 0,3
%A _Gus Wiseman_, Mar 24 2018
%E a(0)=1 prepended by _Alois P. Heinz_, Jan 25 2019
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