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 A301595 Number of thrice-partitions of n. 5
 1, 1, 4, 10, 34, 80, 254, 604, 1785, 4370, 11986, 29286, 80355, 193137, 505952, 1239348, 3181970, 7686199, 19520906, 46931241, 117334784, 282021070, 693721166, 1659075192, 4063164983, 9651686516, 23347635094, 55405326513, 133021397071, 313842472333, 749299686508 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..3244 Gus Wiseman, The a(4) = 34 thrice-partitions of 4. FORMULA O.g.f.: Product_{n > 0} 1/(1 - A063834(n) x^n). EXAMPLE The a(3) = 10 thrice-partitions: ((3)), ((21)), ((111)), ((2)(1)), ((11)(1)), ((1)(1)(1)), ((2))((1)), ((11))((1)), ((1)(1))((1)), ((1))((1))((1)). MAPLE b:= proc(n, i, k) option remember; `if`(n=0 or k=0 or i=1, 1, b(n, i-1, k)+b(i\$2, k-1)*b(n-i, min(n-i, i), k)) end: a:= n-> b(n\$2, 3): seq(a(n), n=0..35); # Alois P. Heinz, Jan 25 2019 MATHEMATICA twie[n_]:=Sum[Times@@PartitionsP/@ptn, {ptn, IntegerPartitions[n]}]; thrie[n_]:=Sum[Times@@twie/@ptn, {ptn, IntegerPartitions[n]}]; Array[thrie, 30] (* Second program: *) b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0 || i == 1, 1, b[n, i - 1, k] + b[i, i, k - 1]*b[n - i, Min[n - i, i], k]]; a[n_] := b[n, n, 3]; a /@ Range[0, 35] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *) CROSSREFS Cf. A000041, A001383, A001970, A061260, A063834, A119442, A196545, A281113, A289501, A300383, A301422, A301462, A301480, A301595, A301598, A301706. Column k=3 of A323718. Sequence in context: A261577 A224217 A066454 * A022445 A091003 A140725 Adjacent sequences: A301592 A301593 A301594 * A301596 A301597 A301598 KEYWORD nonn AUTHOR Gus Wiseman, Mar 24 2018 EXTENSIONS a(0)=1 prepended by Alois P. Heinz, Jan 25 2019 STATUS approved

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Last modified January 30 19:42 EST 2023. Contains 359947 sequences. (Running on oeis4.)