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 A321449 Regular triangle read by rows where T(n,k) is the number of twice-partitions of n with a combined total of k parts. 16
 1, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 4, 5, 5, 0, 1, 4, 8, 8, 7, 0, 1, 6, 13, 19, 16, 11, 0, 1, 6, 17, 27, 32, 24, 15, 0, 1, 8, 24, 47, 61, 62, 41, 22, 0, 1, 8, 30, 63, 99, 111, 100, 61, 30, 0, 1, 10, 38, 94, 158, 209, 210, 170, 95, 42, 0, 1, 10, 45, 119, 229, 328, 382, 348, 259, 136, 56 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS A twice partition of n (A063834) is a choice of an integer partition of each part in an integer partition of n. LINKS Alois P. Heinz, Rows n = 0..200, flattened FORMULA O.g.f.: Product_{n >= 0} 1/(1 - x^n * (Sum_{0 <= k <= n} A008284(n,k) * t^k)). EXAMPLE Triangle begins: 1 0 1 0 1 2 0 1 2 3 0 1 4 5 5 0 1 4 8 8 7 0 1 6 13 19 16 11 0 1 6 17 27 32 24 15 0 1 8 24 47 61 62 41 22 0 1 8 30 63 99 111 100 61 30 The sixth row {0, 1, 6, 13, 19, 16, 11} counts the following twice-partitions: (6) (33) (222) (2211) (21111) (111111) (42) (321) (3111) (1111)(2) (111)(111) (51) (411) (111)(3) (111)(21) (1111)(11) (3)(3) (21)(3) (211)(2) (21)(111) (11111)(1) (4)(2) (22)(2) (21)(21) (211)(11) (11)(11)(11) (5)(1) (31)(2) (22)(11) (2111)(1) (111)(11)(1) (3)(21) (221)(1) (11)(11)(2) (1111)(1)(1) (32)(1) (3)(111) (111)(2)(1) (11)(11)(1)(1) (4)(11) (31)(11) (11)(2)(11) (111)(1)(1)(1) (41)(1) (311)(1) (2)(11)(11) (11)(1)(1)(1)(1) (2)(2)(2) (11)(2)(2) (21)(11)(1) (1)(1)(1)(1)(1)(1) (3)(2)(1) (2)(11)(2) (211)(1)(1) (4)(1)(1) (21)(2)(1) (11)(2)(1)(1) (2)(2)(11) (2)(11)(1)(1) (22)(1)(1) (21)(1)(1)(1) (3)(11)(1) (2)(1)(1)(1)(1) (31)(1)(1) (2)(2)(1)(1) (3)(1)(1)(1) MAPLE g:= proc(n, i) option remember; `if`(n=0 or i=1, x^n, g(n, i-1)+ `if`(i>n, 0, expand(g(n-i, i)*x))) end: b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n, b(n, i-1)+ `if`(i>n, 0, expand(b(n-i, i)*g(i\$2)))) end: T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n\$2)): seq(T(n), n=0..12); # Alois P. Heinz, Nov 11 2018 MATHEMATICA Table[Length[Join@@Table[Select[Tuples[IntegerPartitions/@ptn], Length[Join@@#]==k&], {ptn, IntegerPartitions[n]}]], {n, 0, 10}, {k, 0, n}] (* Second program: *) g[n_, i_] := g[n, i] = If[n == 0 || i == 1, x^n, g[n, i - 1] + If[i > n, 0, Expand[g[n - i, i]*x]]]; b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x^n, b[n, i - 1] + If[i > n, 0, Expand[b[n - i, i]*g[i, i]]]]; T[n_] := CoefficientList[b[n, n], x]; T /@ Range[0, 12] // Flatten (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *) CROSSREFS Row sums are A063834. Last column is A000041. Cf. A000219, A001970, A007716, A008284, A055884, A289501, A317449, A317532, A317533, A320801, A320808. Sequence in context: A089107 A361756 A364912 * A180279 A179968 A323844 Adjacent sequences: A321446 A321447 A321448 * A321450 A321451 A321452 KEYWORD nonn,tabl AUTHOR Gus Wiseman, Nov 10 2018 STATUS approved

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Last modified April 19 18:05 EDT 2024. Contains 371798 sequences. (Running on oeis4.)