Search: a280194 -id:a280194
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A300663
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Expansion of 1/(1 - Sum_{k>=1} mu(k)*x^k), where mu() is the Moebius function (A008683).
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+10
9
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1, 1, 0, -2, -3, -2, 3, 8, 8, -2, -16, -24, -10, 24, 59, 54, -11, -117, -174, -90, 162, 431, 449, -20, -835, -1393, -848, 1062, 3352, 3748, 317, -6257, -11134, -7583, 7294, 25956, 30786, 5217, -46545, -88132, -65062, 48534, 199234, 249263, 63034, -342174, -691679, -554002
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OFFSET
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0,4
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COMMENTS
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LINKS
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FORMULA
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G.f.: 1/(1 - Sum_{k>=1} A008683(k)*x^k).
a(0) = 1; a(n) = Sum_{k=1..n} mu(k) * a(n-k). - Seiichi Manyama, Apr 06 2022
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MAPLE
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a:= proc(n) option remember; `if`(n=0, 1, add(
numtheory[mobius](j)*a(n-j), j=1..n))
end:
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MATHEMATICA
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nmax = 47; CoefficientList[Series[1/(1 - Sum[MoebiusMu[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[MoebiusMu[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 47}]
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PROG
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(PARI) my(N=66, x='x+O('x^N)); Vec(1/(1-sum(k=1, N, moebius(k)*x^k))) \\ Seiichi Manyama, Apr 06 2022
(PARI) a(n) = if(n==0, 1, sum(k=1, n, moebius(k)*a(n-k))); \\ Seiichi Manyama, Apr 06 2022
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A331846
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Number of compositions (ordered partitions) of n into distinct squarefree parts.
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+10
5
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1, 1, 1, 3, 2, 3, 9, 5, 12, 16, 21, 41, 42, 49, 59, 79, 130, 231, 230, 295, 226, 495, 609, 699, 1472, 1042, 1377, 2308, 2982, 3425, 3879, 4877, 7156, 7189, 13531, 14797, 13570, 19551, 27667, 30327, 36382, 47519, 60783, 70561, 78330, 136988, 121659, 174851
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OFFSET
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0,4
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LINKS
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EXAMPLE
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a(7) = 5 because we have [7], [6, 1], [5, 2], [2, 5] and [1, 6].
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CROSSREFS
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Cf. A005117, A032020, A032021, A032022, A087188, A218396, A219107, A280194, A331843, A331844, A331845, A331847.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A347777
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Number of compositions (ordered partitions) of n into at most 2 squarefree parts.
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+10
5
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1, 1, 2, 3, 3, 3, 4, 5, 6, 4, 4, 5, 7, 7, 6, 7, 10, 9, 8, 7, 11, 9, 10, 9, 14, 10, 10, 10, 13, 11, 10, 11, 16, 13, 14, 13, 22, 15, 14, 15, 22, 17, 16, 19, 25, 20, 16, 17, 26, 20, 16, 15, 27, 21, 20, 15, 26, 21, 22, 19, 29, 23, 22, 22, 30, 23, 22, 23, 35, 25, 26
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OFFSET
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0,3
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LINKS
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MATHEMATICA
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Table[Length@Flatten[Permutations/@IntegerPartitions[n, 2, Select[Range@n, SquareFreeQ]], 1], {n, 0, 100}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A347778
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Number of compositions (ordered partitions) of n into at most 3 squarefree parts.
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+10
5
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1, 1, 2, 4, 6, 9, 11, 14, 18, 23, 25, 26, 28, 37, 42, 44, 46, 57, 66, 70, 68, 79, 88, 96, 92, 106, 115, 124, 118, 134, 143, 149, 142, 161, 176, 187, 178, 210, 221, 235, 214, 251, 266, 280, 262, 300, 328, 335, 308, 350, 379, 385, 342, 396, 425, 447, 392, 442, 475
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OFFSET
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0,3
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LINKS
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MATHEMATICA
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Table[Length@Flatten[Permutations/@IntegerPartitions[n, 3, Select[Range@n, SquareFreeQ]], 1], {n, 0, 58}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A347779
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Number of compositions (ordered partitions) of n into at most 4 squarefree parts.
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+10
5
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1, 1, 2, 4, 7, 13, 21, 30, 41, 55, 75, 94, 111, 129, 158, 192, 224, 249, 290, 346, 403, 439, 488, 556, 639, 686, 749, 828, 939, 1002, 1081, 1173, 1304, 1373, 1464, 1579, 1750, 1838, 1963, 2111, 2337, 2423, 2574, 2740, 3023, 3120, 3292, 3511, 3858, 3978, 4157, 4413
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OFFSET
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0,3
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LINKS
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MATHEMATICA
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Table[Length@Flatten[Permutations/@IntegerPartitions[n, 4, Select[Range@n, SquareFreeQ]], 1], {n, 0, 51}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A300706
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Number of compositions (ordered partitions) of n into squarefree parts that do not divide n.
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+10
4
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1, 0, 0, 0, 0, 2, 0, 5, 2, 5, 2, 27, 2, 67, 12, 16, 28, 366, 4, 848, 28, 182, 153, 4591, 20, 4172, 554, 2217, 558, 57695, 6, 134118, 3834, 14629, 6972, 97478, 258, 1684852, 24467, 120869, 5308, 9104710, 189, 21165023, 124427, 117017, 297830, 114373157, 3394, 126979537, 72158, 7655405
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OFFSET
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0,6
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LINKS
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EXAMPLE
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a(18) = 4 because we have [13, 5], [11, 7], [7, 11] and [5, 13].
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MAPLE
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with(numtheory):
a:= proc(m) option remember; local b; b:= proc(n) option
remember; `if`(n=0, 1, add(`if`(not issqrfree(j) or
irem(m, j)=0, 0, b(n-j)), j=2..n)) end; b(m)
end:
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MATHEMATICA
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Table[SeriesCoefficient[1/(1 - Sum[Boole[Mod[n, k] != 0 && SquareFreeQ[k]] x^k, {k, 1, n}]), {x, 0, n}], {n, 0, 51}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A347780
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Number of compositions (ordered partitions) of n into at most 5 squarefree parts.
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+10
4
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1, 1, 2, 4, 7, 14, 26, 45, 71, 105, 151, 214, 291, 379, 473, 593, 744, 919, 1095, 1301, 1563, 1884, 2203, 2536, 2929, 3427, 3929, 4433, 4979, 5692, 6422, 7158, 7904, 8863, 9844, 10830, 11810, 13078, 14378, 15706, 17007, 18718, 20424, 22165, 23803, 26025
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OFFSET
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0,3
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LINKS
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MATHEMATICA
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Table[Length@Flatten[Permutations/@IntegerPartitions[n, 5, Select[Range@n, SquareFreeQ]], 1], {n, 0, 45}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A284464
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Number of compositions (ordered partitions) of n into squarefree divisors of n.
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+10
3
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1, 1, 2, 2, 5, 2, 25, 2, 34, 19, 129, 2, 1046, 2, 742, 450, 1597, 2, 44254, 2, 27517, 3321, 29967, 2, 1872757, 571, 200390, 18560, 854850, 2, 154004511, 2, 3524578, 226020, 9262157, 51886, 3353855285, 2, 63346598, 2044895, 1255304727, 2, 185493291001, 2, 1282451595, 345852035, 2972038875, 2, 6006303471178
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = [x^n] 1/(1 - Sum_{d|n, |mu(d)| = 1} x^d), where mu(d) is the Moebius function (A008683).
a(n) = 2 if n is a prime.
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EXAMPLE
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a(4) = 5 because 4 has 3 divisors {1, 2, 4} among which 2 are squarefree {1, 2} therefore we have [2, 2], [2, 1, 1], [1, 2, 1], [1, 2, 2] and [1, 1, 1, 1].
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MAPLE
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with(numtheory):
a:= proc(n) option remember; local b, l;
l, b:= select(issqrfree, divisors(n)),
proc(m) option remember; `if`(m=0, 1,
add(`if`(j>m, 0, b(m-j)), j=l))
end; b(n)
end:
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MATHEMATICA
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Table[d = Divisors[n]; Coefficient[Series[1/(1 - Sum[MoebiusMu[d[[k]]]^2 x^d[[k]], {k, Length[d]}]), {x, 0, n}], x, n], {n, 0, 48}]
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PROG
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(Python)
from sympy import divisors
from sympy.ntheory.factor_ import core
from sympy.core.cache import cacheit
@cacheit
def a(n):
l=[x for x in divisors(n) if core(x)==x]
@cacheit
def b(m): return 1 if m==0 else sum(b(m - j) for j in l if j <= m)
return b(n)
print([a(n) for n in range(51)]) # Indranil Ghosh, Aug 01 2017, after Maple code
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A280197
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Expansion of 1/(1 - Sum_{k>=2} mu(k)^2*x^k), where mu(k) is the Moebius function (A008683).
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+10
1
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1, 0, 1, 1, 1, 3, 3, 6, 8, 12, 20, 28, 45, 68, 102, 159, 238, 367, 557, 849, 1298, 1973, 3015, 4592, 7002, 10679, 16276, 24822, 37841, 57696, 87971, 134119, 204497, 311783, 475370, 724786, 1105053, 1684853, 2568837, 3916642, 5971587, 9104711, 13881698, 21165024, 32269721, 49200718, 75014949, 114373158, 174381511
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OFFSET
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0,6
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COMMENTS
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Number of compositions (ordered partitions) into squarefree parts > 1 (A144338).
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LINKS
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FORMULA
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G.f.: 1/(1 - Sum_{k>=2} mu(k)^2*x^k).
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EXAMPLE
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a(5) = 3 because we have [5], [3, 2] and [2, 3].
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MAPLE
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N:= 100: # for a(0)..a(N)
g:= 1/(1-add(numtheory:-mobius(k)^2*x^k, k=2..N)):
S:= series(g, x, N+1):
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MATHEMATICA
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nmax = 48; CoefficientList[Series[1/(1 - Sum[MoebiusMu[k]^2 x^k, {k, 2, nmax}]), {x, 0, nmax}], x]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A280198
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Expansion of 1/(1 - Sum_{k>=1} mu(2*k-1)^2*x^(2*k-1)), where mu() is the Moebius function (A008683).
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+10
1
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1, 1, 1, 2, 3, 5, 8, 13, 21, 33, 53, 86, 138, 222, 357, 574, 923, 1484, 2387, 3839, 6173, 9927, 15964, 25672, 41284, 66389, 106762, 171686, 276091, 443989, 713988, 1148179, 1846411, 2969252, 4774918, 7678647, 12348195, 19857396, 31933099, 51352294, 82580715, 132799801, 213558181, 343427445, 552272966, 888121883, 1428207656
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OFFSET
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0,4
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COMMENTS
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Number of compositions (ordered partitions) into odd squarefree parts (A056911).
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LINKS
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FORMULA
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G.f.: 1/(1 - Sum_{k>=1} mu(2*k-1)^2*x^(2*k-1)).
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EXAMPLE
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a(4) = 3 because we have [3, 1], [1, 3] and [1, 1, 1, 1].
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MATHEMATICA
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nmax = 46; CoefficientList[Series[1/(1 - Sum[MoebiusMu[2 k - 1]^2 x^(2 k - 1), {k, 1, nmax}]), {x, 0, nmax}], x]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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