A290529 - OEIS

A290529

Inverse of the factorization matrix in the Lambert series factorization theorem.

%I #20 Nov 18 2018 09:56:08

%S 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,

%T 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,

%U 5,3,2,1,1,47,36,26,19,14,10,7,5,3,2,1,1

%N Inverse of the factorization matrix in the Lambert series factorization theorem.

%C 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),

%C 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.

%C In particular, for a fixed arithmetic function f, we may expand its Lambert series generating function by the factorization

%C 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

%C 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.

%H M. Merca, <a href="https://doi.org/10.1007/s11139-016-9856-3">The Lambert series factorization theorem</a>, Ramanujan J. (2017).

%H M. Merca and M. D. Schmidt, <a href="https://arxiv.org/abs/1706.00393">Generating special arithmetic functions by Lambert series factorizations</a>, arXiv:1706.00393 [math.NT], 2017.

%H M. D. Schmidt, <a href="https://arxiv.org/abs/1701.06257">New recurrence relations and matrix equations for arithmetic functions generated by Lambert series</a>, arXiv:1701.06257 [math.NT], Acta Arith. (2017).

%F Möbius transform of the shifted partition numbers (A000041), p(n-k).

%F An explicit formula for the triangular sequence is given by:

%F 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).

%F The first column of the sequence is A133732.

%F The sequence s_(n,k)^(-1) - p(n-k) is expanded as the following similarly shaped triangle:

%F 0;

%F -1, 0;

%F -1, 0, 0;

%F -1, -1, 0, 0;

%F -1, 0, 0, 0, 0;

%F -2, -2, -1, 0, 0, 0;

%F -1, 0, 0, 0, 0, 0, 0;

%F -3, -2, -1, -1, 0, 0, 0, 0;

%F -2, -1, -1, 0, 0, 0, 0, 0, 0;

%F -5, -4, -2, -1, -1, 0, 0, 0, 0, 0;

%F -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

%F -9, -6, -4, -3, -1, -1, 0, 0, 0, 0, 0, 0;

%F (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).

%F 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.

%F The similarly shaped triangle denoting the inverse matrix of the sequence is given by:

%F 1;

%F 0, 1;

%F -1, -1, 1;

%F -1, 0, -1, 1;

%F -1, -1, -1, -1, 1;

%F 0, 0, 1, -1, -1, 1;

%F 0, 0, -1, 0, -1, -1, 1;

%F 1, 0, 0, 1, 0, -1, -1, 1;

%F 1, 1, 1, 0, 0, 0, -1, -1, 1;

%F 1, 0, 0, -1, 2, 0, 0, -1, -1, 1;

%F 1, 1, 0, 1, -1, 1, 0, 0, -1, -1, 1;

%F 1, 0, 1, 1, 0, 1, 1, 0, 0, -1, -1, 1.

%e Triangle begins:

%e 1,

%e 0, 1,

%e 1, 1, 1,

%e 2, 1, 1, 1,

%e 4, 3, 2, 1, 1,

%e 5, 3, 2, 2, 1, 1,

%e 10, 7, 5, 3, 2, 1, 1,

%e ...

%t (* View as a table *)

%t Table[DivisorSum[n, PartitionsP[# - k] MoebiusMu[n/#] &], {n, 1, 12}, {k, 1, n}] // TableForm

%t (* Flattened sequence entry as listed here *)

%t Table[DivisorSum[n, PartitionsP[# - k] MoebiusMu[n/#] &], {n, 1, 12}, {k, 1, n}] // Flatten

%t (* Compare with its inverse matrix *)

%t Table[DivisorSum[n, PartitionsP[# - k] MoebiusMu[n/#] &], {n, 1, 12}, {k, 1, 12}] // Inverse // MatrixForm

%t Table[SeriesCoefficient[(q^k)/(1-q^k) QPochhammer[q, q], {q, 0, n}], {n, 1, 12}, {k, 1, 12}] // MatrixForm

%t (* Remove dominant partition function terms in the sequence *)

%t Table[DivisorSum[n, PartitionsP[# - k] MoebiusMu[n/#] &]-PartitionsP[n-k], {n, 1, 12}, {k, 1, n}] // TableForm

%o (PARI) T(n, k) = sumdiv(n, d, numbpart(d-k)*moebius(n/d));

%o tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n, k), ", ")); print); \\ _Michel Marcus_, Nov 17 2018

%Y Cf. A133732, A000041, A078616 (first column of the inverse sequence).

%K nonn,tabl

%O 1,7

%A _Maxie D. Schmidt_, Aug 04 2017