A337009 Triangle of the Multiset Transform of the Fibonacci Sequence.

1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 5, 3, 1, 1, 8, 11, 6, 3, 1, 1, 13, 19, 13, 6, 3, 1, 1, 21, 37, 25, 14, 6, 3, 1, 1, 34, 65, 52, 27, 14, 6, 3, 1, 1, 55, 120, 98, 58, 28, 14, 6, 3, 1, 1, 89, 210, 191, 113, 60, 28, 14, 6, 3, 1, 1, 144, 376, 360, 229, 119, 61, 28, 14, 6, 3, 1, 1, 233, 654, 678, 443, 244, 121, 61, 28, 14, 6, 3, 1, 1

Offset: 1

Author

R. J. Mathar, Aug 11 2020

Keywords

• nonn
• tabl

Comments

Short definition of the Multiset Transformation: supposed we have F(w) distinct objects of weight w. Then T(n,k) is the number of bags of objects with total weight n containing k objects. Multisets means that objects may appear more than once in the bag, but the order of the objects in the bag does not matter.

Apparently A200544 is the limit of the reversed rows as n approaches infinity.

Examples

			The triangle starts with rows n>=1 and columns k>=1:
    1
    1     1
    2     1     1
    3     3     1     1
    5     5     3     1     1
    8    11     6     3     1     1
   13    19    13     6     3     1     1
   21    37    25    14     6     3     1     1
   34    65    52    27    14     6     3     1     1
   55   120    98    58    28    14     6     3     1     1
   89   210   191   113    60    28    14     6     3     1     1
  144   376   360   229   119    61    28    14     6     3     1     1
  233   654   678   443   244   121    61    28    14     6     3     1     1
  377  1149  1255   866   481   250   122    61    28    14     6     3     1  1
  ...
		

Links

• Alois P. Heinz, Rows n = 1..200, flattened
• Index to Sequence pairs related by Multiset transformations

Crossrefs

Cf. A000045 (column k=1), A089098 (column k=2), A166861 (row sums), A200544 (limiting row?), A357475.

Programs

Maple

F:= proc(n) option remember; (<<1|1>, <1|0>>^n)[1, 2] end:
b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, x^n,
      add(binomial(F(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
seq(T(n), n=1..12);  # Alois P. Heinz, Oct 29 2021

Mathematica

nn = 13;
Rest@CoefficientList[#, y]& /@ (Series[Product[1/(1 - y x^i)^Fibonacci[i], {i, 1, nn}], {x, 0, nn}] // Rest@CoefficientList[#, x]&) // Flatten (* Jean-François Alcover, Oct 29 2021 *)

Formula

G.f.: Product_{j>=1} 1/(1-y*x^j)^Fibonacci(j). - Jean-François Alcover, Oct 29 2021

Sum_{k=0..n} (-1)^k * T(n,k) = A357475(n). - Alois P. Heinz, Apr 30 2023