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

0,3

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.

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A000041, A000045.

Sequence in context: A327288 A102688 A236559 * A341581 A001629 A159230

Adjacent sequences: A275385 A275386 A275387 * A275389 A275390 A275391

Keyword

nonn

Author

Vladimir Reshetnikov, Sep 26 2016

Status

approved