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.

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

Offset: 0

Author

Gus Wiseman, Nov 10 2018

Keywords

• nonn
• tabl

Comments

A twice partition of n (A063834) is a choice of an integer partition of each part in an integer partition of n.

Examples

			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)
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened

Crossrefs

Row sums are A063834. Last column is A000041.

Cf. A000219, A001970, A007716, A008284, A055884, A289501, A317449, A317532, A317533, A320801, A320808.

Programs

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 *)

Formula

O.g.f.: Product_{n >= 0} 1/(1 - x^n * (Sum_{0 <= k <= n} A008284(n,k) * t^k)).