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