cp's OEIS Frontend

A337009 Triangle of the Multiset Transform of the Fibonacci Sequence.

%I A337009 #19 Apr 30 2023 15:55:01
%S A337009 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,
%T A337009 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,
%U A337009 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
%N A337009 Triangle of the Multiset Transform of the Fibonacci Sequence.
%C A337009 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.
%C A337009 Apparently A200544 is the limit of the reversed rows as n approaches infinity.
%H A337009 Alois P. Heinz, <a href="/A337009/b337009.txt">Rows n = 1..200, flattened</a>
%H A337009 <a href="/index/Mu#multiplicative_completely">Index to Sequence pairs related by Multiset transformations</a>
%F A337009 G.f.: Product_{j>=1} 1/(1-y*x^j)^Fibonacci(j). - _Jean-François Alcover_, Oct 29 2021
%F A337009 Sum_{k=0..n} (-1)^k * T(n,k) = A357475(n). - _Alois P. Heinz_, Apr 30 2023
%e A337009 The triangle starts with rows n>=1 and columns k>=1:
%e A337009     1
%e A337009     1     1
%e A337009     2     1     1
%e A337009     3     3     1     1
%e A337009     5     5     3     1     1
%e A337009     8    11     6     3     1     1
%e A337009    13    19    13     6     3     1     1
%e A337009    21    37    25    14     6     3     1     1
%e A337009    34    65    52    27    14     6     3     1     1
%e A337009    55   120    98    58    28    14     6     3     1     1
%e A337009    89   210   191   113    60    28    14     6     3     1     1
%e A337009   144   376   360   229   119    61    28    14     6     3     1     1
%e A337009   233   654   678   443   244   121    61    28    14     6     3     1     1
%e A337009   377  1149  1255   866   481   250   122    61    28    14     6     3     1  1
%e A337009   ...
%p A337009 F:= proc(n) option remember; (<<1|1>, <1|0>>^n)[1, 2] end:
%p A337009 b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, x^n,
%p A337009       add(binomial(F(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)))
%p A337009     end:
%p A337009 T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
%p A337009 seq(T(n), n=1..12);  # _Alois P. Heinz_, Oct 29 2021
%t A337009 nn = 13;
%t A337009 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 *)
%Y A337009 Cf. A000045 (column k=1), A089098 (column k=2), A166861 (row sums), A200544 (limiting row?), A357475.
%K A337009 nonn,tabl
%O A337009 1,4
%A A337009 _R. J. Mathar_, Aug 11 2020