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

1, 3, 6, 11, 18, 29, 42, 62, 86, 119, 159, 211, 273, 352, 446, 562, 697, 864, 1054, 1284, 1550, 1860, 2220, 2639, 3114, 3669, 4293, 5011, 5823, 6745, 7783, 8956, 10268, 11747, 13390, 15237, 17281, 19561, 22089, 24889, 27979, 31405, 35157, 39309, 43856, 48849, 54319, 60309, 66840, 73992, 81760, 90243

Offset: 1

Author

Ilya Gutkovskiy, Jan 27 2017

Keywords

• nonn

Comments

Total number of parts in all partitions of n into Fibonacci parts (with a single type of 1).

Convolution of A003107 and A005086.

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000
• Index entries for related partition-counting sequences

Crossrefs

Cf. A000045, A003107, A005086, A240225, A319394.

Programs

Maple

h:= proc(n) option remember; `if`(n<1, 0, `if`((t->
      issqr(t+4) or issqr(t-4))(5*n^2), n, h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n],
       b(n, h(i-1))+(p->p+[0, p[1]])(b(n-i, h(min(n-i, i)))))
    end:
a:= n-> b(n, h(n))[2]:
seq(a(n), n=1..70);  # Alois P. Heinz, Sep 18 2018

Mathematica

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]]

Formula

G.f.: Sum_{k>=2} x^Fibonacci(k)/(1 - x^Fibonacci(k)) / Product_{k>=2} (1 - x^Fibonacci(k)).

a(n) = Sum_{k=1..n} k * A319394(n,k). - Alois P. Heinz, Sep 18 2018