A321460 -id:A321460 - cp's OEIS frontend

A066815 Number of partitions of n into sums of products.

1, 1, 2, 3, 6, 8, 14, 19, 33, 45, 69, 94, 148, 197, 289, 390, 575, 762, 1086, 1439, 2040, 2687, 3712, 4874, 6749, 8792, 11918, 15526, 20998, 27164, 36277, 46820, 62367, 80146, 105569, 135326, 177979, 227139, 296027, 377142, 490554, 622526, 804158

Offset: 0

Vladeta Jovovic, Jan 20 2002

nonn

Number of ways to choose a factorization of each part of an integer partition of n. - Gus Wiseman, Sep 05 2018

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1, g(n) = A001055(n). - Seiichi Manyama, Nov 14 2018

			From _Gus Wiseman_, Sep 05 2018: (Start)
The a(6) = 14 partitions of 6 into sums of products:
  6, 2*3,
  5+1, 4+2, 2*2+2, 3+3,
  4+1+1, 2*2+1+1, 3+2+1, 2+2+2,
  3+1+1+1, 2+2+1+1,
  2+1+1+1+1,
  1+1+1+1+1+1.
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000041, A001055, A066739, A321460.

Cf. A001970, A050336, A063834, A065026, A281113, A284639, A318948, A318949.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
Table[Length[Join@@Table[Tuples[facs/@ptn],{ptn,IntegerPartitions[n]}]],{n,20}] (* Gus Wiseman, Sep 05 2018 *)

G.f.: Product_{k>=1} 1/(1-A001055(k)*x^k).

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

Renamed by T. D. Noe, May 24 2011

A066806 Expansion of Product_{k>=1} (1+x^k)^A001055(k).

Original entry on oeis.org

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

Offset: 0

Vladeta Jovovic, Jan 19 2002

nonn

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

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

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

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-François Alcover, Mar 23 2017, after Alois P. Heinz *)

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 range(1, n + 1))//n
print([a(n) for n in range(61)]) # Indranil Ghosh, Aug 19 2017, after Maple code

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).

A319517 Expansion of Product_{1 <= i_1 <= i_2 <= i_3 <= i_4} (1 - x^(i_1i_2i_3*i_4)).

Original entry on oeis.org

1, -1, -1, 0, -1, 2, 0, 2, -1, 0, 3, -1, -2, -1, 1, -6, -1, 0, 0, 0, 7, -1, 1, -2, 4, 1, -2, 11, 1, -2, -10, 11, -11, 15, -16, -6, -7, -10, -1, 10, -5, -10, 12, -20, 19, -16, 24, -2, 28, -9, 41, -6, 15, 20, 4, -21, -15, -13, -14, 13, -73, 67, -30, -44, -19, 31, -30

Offset: 0

Seiichi Manyama, Nov 14 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Convolution inverse of A321566.
Product_{1 <= i_1 <= i_2 <= ... <= i_b} (1 - x^(i_1 * i_2 * ... * i_b)): A010815 (b=1), A321299 (b=2), A321361 (b=3), this sequence (b=4).
Cf. A218320, A321460, A321567.

G.f.: Product_{k>0} (1 - x^k)^A218320(k).

A321594 Expansion of Product_{k>0} (1 - A001055(k)*x^k).

Original entry on oeis.org

1, -1, -1, 0, -1, 2, 0, 2, -2, 1, 4, -1, -4, 0, 4, -8, -3, 0, 0, -2, 17, -8, -2, -11, 4, 7, 0, 22, 26, -30, -32, 30, -18, 57, -58, 28, -12, -28, -41, 97, -11, -36, 8, -95, -5, -69, -29, 104, 76, 14, 209, -145, 29, 46, 371, -437, 0, -336, 116, -94, -388, 952, 449, -665

Offset: 0

Seiichi Manyama, Nov 14 2018

sign

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = A001055(n).

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Convolution inverse of A066815.

Cf. A001055, A066816, A321460.

cp's OEIS Frontend

A066815 Number of partitions of n into sums of products.

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A066806 Expansion of Product_{k>=1} (1+x^k)^A001055(k).

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A319517 Expansion of Product_{1 <= i_1 <= i_2 <= i_3 <= i_4} (1 - x^(i_1i_2i_3*i_4)).

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A321594 Expansion of Product_{k>0} (1 - A001055(k)*x^k).

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cp's OEIS Frontend

A066815 Number of partitions of n into sums of products.

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A066806 Expansion of Product_{k>=1} (1+x^k)^A001055(k).

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Maple

Mathematica

Python

Formula

A319517 Expansion of Product_{1 <= i_1 <= i_2 <= i_3 <= i_4} (1 - x^(i_1*i_2*i_3*i_4)).

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A321594 Expansion of Product_{k>0} (1 - A001055(k)*x^k).

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A319517 Expansion of Product_{1 <= i_1 <= i_2 <= i_3 <= i_4} (1 - x^(i_1i_2i_3*i_4)).