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A290529
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Inverse of the factorization matrix in the Lambert series factorization theorem.
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0
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1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 4, 3, 2, 1, 1, 5, 3, 2, 2, 1, 1, 10, 7, 5, 3, 2, 1, 1, 12, 9, 6, 4, 3, 2, 1, 1, 20, 14, 10, 7, 5, 3, 2, 1, 1, 25, 18, 13, 10, 6, 5, 3, 2, 1, 1, 41, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1, 47, 36, 26, 19, 14, 10, 7, 5, 3, 2, 1, 1
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OFFSET
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1,7
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COMMENTS
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The entries in the inverse matrix for the sequence s_(n,k) := [q^n] (q^k) / (1-q^k) (q; q)_inf = s_o(n, k) - s_e(n, k),
which is the difference of the number of k's in all partitions of n into an odd (even) number of distinct parts. The sequence arises in identities related to _the_ original formulation of the Lambert series factorization theorem in Merca's article in Ramanujan J. below.
In particular, for a fixed arithmetic function f, we may expand its Lambert series generating function by the factorization
Sum_{n>=1} f(n)*q^n/(1-q^n) = (1/(q; q)_inf) Sum_{n>=1} Sum_{k=1..n} s_(n,k) f(k) q^n, and then by inversion by this sequence, denoted s_(n,k)^(-1), we have the corresponding identity that
f(n) = Sum_{k=1..n} s_(n,k)^(-1) Sum_{j: G_j <= k} (-1)^ceiling(j/2) (f * 1)(k - G_j), where G_j denotes the sequence of interleaved pentagonal numbers (A001318) for j >= 1.
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LINKS
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FORMULA
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Möbius transform of the shifted partition numbers (A000041), p(n-k).
An explicit formula for the triangular sequence is given by:
s_(n,k)^(-1) = Sum_{d|n} p(d-k)*Mobius(n/d), where p(n) is Euler's partition function (A000041) and Mobius(n) is the Mobius function (A008683).
The first column of the sequence is A133732.
The sequence s_(n,k)^(-1) - p(n-k) is expanded as the following similarly shaped triangle:
0;
-1, 0;
-1, 0, 0;
-1, -1, 0, 0;
-1, 0, 0, 0, 0;
-2, -2, -1, 0, 0, 0;
-1, 0, 0, 0, 0, 0, 0;
-3, -2, -1, -1, 0, 0, 0, 0;
-2, -1, -1, 0, 0, 0, 0, 0, 0;
-5, -4, -2, -1, -1, 0, 0, 0, 0, 0;
-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
-9, -6, -4, -3, -1, -1, 0, 0, 0, 0, 0, 0;
(in other words, the nonzero terms in the triangle after subtracting off p(n-k) are comparatively sparse at only about half of the remaining entries).
The Lambert series generating function for the sequence at a fixed k >= 1 is given by: Sum_{n >= 1} s_(n,k)^(-1) q^n/(1-q^n) = q^k / (q; q)_inf.
The similarly shaped triangle denoting the inverse matrix of the sequence is given by:
1;
0, 1;
-1, -1, 1;
-1, 0, -1, 1;
-1, -1, -1, -1, 1;
0, 0, 1, -1, -1, 1;
0, 0, -1, 0, -1, -1, 1;
1, 0, 0, 1, 0, -1, -1, 1;
1, 1, 1, 0, 0, 0, -1, -1, 1;
1, 0, 0, -1, 2, 0, 0, -1, -1, 1;
1, 1, 0, 1, -1, 1, 0, 0, -1, -1, 1;
1, 0, 1, 1, 0, 1, 1, 0, 0, -1, -1, 1.
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EXAMPLE
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Triangle begins:
1,
0, 1,
1, 1, 1,
2, 1, 1, 1,
4, 3, 2, 1, 1,
5, 3, 2, 2, 1, 1,
10, 7, 5, 3, 2, 1, 1,
...
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MATHEMATICA
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(* View as a table *)
Table[DivisorSum[n, PartitionsP[# - k] MoebiusMu[n/#] &], {n, 1, 12}, {k, 1, n}] // TableForm
(* Flattened sequence entry as listed here *)
Table[DivisorSum[n, PartitionsP[# - k] MoebiusMu[n/#] &], {n, 1, 12}, {k, 1, n}] // Flatten
(* Compare with its inverse matrix *)
Table[DivisorSum[n, PartitionsP[# - k] MoebiusMu[n/#] &], {n, 1, 12}, {k, 1, 12}] // Inverse // MatrixForm
Table[SeriesCoefficient[(q^k)/(1-q^k) QPochhammer[q, q], {q, 0, n}], {n, 1, 12}, {k, 1, 12}] // MatrixForm
(* Remove dominant partition function terms in the sequence *)
Table[DivisorSum[n, PartitionsP[# - k] MoebiusMu[n/#] &]-PartitionsP[n-k], {n, 1, 12}, {k, 1, n}] // TableForm
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PROG
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(PARI) T(n, k) = sumdiv(n, d, numbpart(d-k)*moebius(n/d));
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Nov 17 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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