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A066806 Expansion of Product_{k>=1} (1+x^k)^A001055(k). 5
1, 1, 1, 2, 3, 4, 6, 8, 11, 16, 21, 27, 38, 49, 63, 84, 109, 138, 180, 228, 289, 369, 463, 578, 732, 911, 1128, 1407, 1741, 2140, 2646, 3243, 3968, 4862, 5925, 7198, 8770, 10620, 12833, 15524, 18718, 22502, 27075, 32467, 38873, 46537, 55565, 66220, 78946 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

FORMULA

a(n) = 1/n*Sum_{k=1..n} a(n-k)*b(k), n>0, a(0)=1, b(k)=Sum_{d|k} (-1)^(n/d+1)*d*A001055(d).

MAPLE

with(numtheory):

g:= proc(n, k) option remember; `if`(n>k, 0, 1)+

      `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d)),

         d=divisors(n) minus {1, n}))

    end:

b:= proc(n) b(n):= add((-1)^(n/d+1)*d*g(d$2), d=divisors(n)) end:

a:= proc(n) a(n):= `if`(n=0, 1, add(a(n-k)*b(k), k=1..n)/n) end:

seq(a(n), n=0..60);  # Alois P. Heinz, May 16 2014

MATHEMATICA

g[n_, k_] := g[n, k] = If[n > k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d > k, 0, g[n/d, d]], {d, Divisors[n] ~Complement~ {1, n}}]];

b[n_] := b[n] = Sum[(-1)^(n/d + 1)*d*g[d, d], {d, Divisors[n]}];

a[n_] := a[n] = If[n == 0, 1, Sum[a[n - k]*b[k], {k, 1, n}]/n];

Table[a[n], {n, 0, 60}] (* Jean-Fran├žois Alcover, Mar 23 2017, after Alois P. Heinz *)

PROG

(Python)

from sympy.core.cache import cacheit

from sympy import divisors, isprime

@cacheit

def g(n, k): return (0 if n>k else 1) + (0 if isprime(n) else sum([0 if d>k else g(n/d, d) for d in divisors(n)[1:-1]]))

@cacheit

def b(n): return sum([(-1)**(n/d + 1)*d*g(d, d) for d in divisors(n)])

@cacheit

def a(n): return 1 if n==0 else sum([a(n - k)*b(k) for k in xrange(1, n + 1)])/n

print map(a, xrange(61)) # Indranil Ghosh, Aug 19 2017, after Maple code

CROSSREFS

Cf. A000041, A001055, A066739, A057567, A321460.

Sequence in context: A064660 A321359 A321567 * A271486 A226187 A294922

Adjacent sequences:  A066803 A066804 A066805 * A066807 A066808 A066809

KEYWORD

nonn

AUTHOR

Vladeta Jovovic, Jan 19 2002

STATUS

approved

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Last modified April 18 12:34 EDT 2019. Contains 322209 sequences. (Running on oeis4.)