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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
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.
Sequence in context: A089107 A361756 A364912 * A180279 A179968 A323844
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.)