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