A275388 Convolution of Fibonacci numbers (A000045) and partition numbers (A000041).

0, 1, 2, 5, 10, 20, 37, 68, 120, 210, 360, 612, 1028, 1717, 2846, 4698, 7720, 12649, 20666, 33700, 54856, 89183, 144831, 235016, 381102, 617693, 1000753, 1620882, 2624645, 4249245, 6878455, 11133304, 18018601, 29160254, 47188998, 76361562, 123565443, 199944982

Offset: 0

Author

Vladimir Reshetnikov, Sep 26 2016

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Fibonacci Number, Partition Function P, q-Pochhammer Symbol.

Crossrefs

Cf. A000041, A000045.

Programs

Mathematica

Table[Sum[Fibonacci[k] PartitionsP[n - k], {k, 1, n}], {n, 0, 30}]

PARI

a(n)=sum(k=1, n, fibonacci(k)*numbpart(n - k)); \\ Indranil Ghosh, Jun 29 2017

Python

from sympy import fibonacci, npartitions
def a(n): return sum([fibonacci(k)*npartitions(n - k) for k in range(1, n + 1)])
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 29 2017

Formula

a(n) = Sum_{k=1..n} A000045(k)*A000041(n-k).

G.f.: x/((1 - x - x^2) * (x; x)_inf), where (x; x)_inf is the q-Pochhammer symbol.

a(n+1) - a(n) - a(n-1) = A000041(n).

a(n) ~ phi^n / (sqrt(5) * QPochhammer(1/phi)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 27 2016