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

1,4

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.

Links

Alois P. Heinz, Rows n = 1..200, flattened

Index to Sequence pairs related by Multiset transformations

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

Example

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
  ...

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 *)

Crossrefs

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

Sequence in context: A167040 A054450 A344610 * A174802 A238346 A053538

Adjacent sequences: A337006 A337007 A337008 * A337010 A337011 A337012

Keyword

nonn,tabl

Author

R. J. Mathar, Aug 11 2020

Status

approved