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A275388 Convolution of Fibonacci numbers (A000045) and partition numbers (A000041). 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified April 21 17:04 EDT 2021. Contains 343156 sequences. (Running on oeis4.)