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A281689 Expansion of Sum_{k>=2} x^Fibonacci(k)/(1 - x^Fibonacci(k)) / Product_{k>=2} (1 - x^Fibonacci(k)).

%I A281689 #7 Sep 18 2018 16:10:05
%S A281689 1,3,6,11,18,29,42,62,86,119,159,211,273,352,446,562,697,864,1054,
%T A281689 1284,1550,1860,2220,2639,3114,3669,4293,5011,5823,6745,7783,8956,
%U A281689 10268,11747,13390,15237,17281,19561,22089,24889,27979,31405,35157,39309,43856,48849,54319,60309,66840,73992,81760,90243
%N A281689 Expansion of Sum_{k>=2} x^Fibonacci(k)/(1 - x^Fibonacci(k)) / Product_{k>=2} (1 - x^Fibonacci(k)).
%C A281689 Total number of parts in all partitions of n into Fibonacci parts (with a single type of 1).
%C A281689 Convolution of A003107 and A005086.
%H A281689 Alois P. Heinz, <a href="/A281689/b281689.txt">Table of n, a(n) for n = 1..20000</a>
%H A281689 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F A281689 G.f.: Sum_{k>=2} x^Fibonacci(k)/(1 - x^Fibonacci(k)) / Product_{k>=2} (1 - x^Fibonacci(k)).
%F A281689 a(n) = Sum_{k=1..n} k * A319394(n,k). - _Alois P. Heinz_, Sep 18 2018
%e A281689 a(5) = 18 because we have [5], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1] and 1 + 2 + 3 + 3 + 4 + 5 = 18.
%p A281689 h:= proc(n) option remember; `if`(n<1, 0, `if`((t->
%p A281689       issqr(t+4) or issqr(t-4))(5*n^2), n, h(n-1)))
%p A281689     end:
%p A281689 b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n],
%p A281689        b(n, h(i-1))+(p->p+[0, p[1]])(b(n-i, h(min(n-i, i)))))
%p A281689     end:
%p A281689 a:= n-> b(n, h(n))[2]:
%p A281689 seq(a(n), n=1..70);  # _Alois P. Heinz_, Sep 18 2018
%t A281689 Rest[CoefficientList[Series[Sum[x^Fibonacci[k]/(1 - x^Fibonacci[k]), {k, 2, 20}]/Product[1 - x^Fibonacci[k], {k, 2, 20}], {x, 0, 52}], x]]
%Y A281689 Cf. A000045, A003107, A005086, A240225, A319394.
%K A281689 nonn
%O A281689 1,2
%A A281689 _Ilya Gutkovskiy_, Jan 27 2017