A290529 Inverse of the factorization matrix in the Lambert series factorization theorem.

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

Offset

1,7

Comments

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.

Links

Table of n, a(n) for n=1..78.

M. Merca, The Lambert series factorization theorem, Ramanujan J. (2017).

M. Merca and M. D. Schmidt, Generating special arithmetic functions by Lambert series factorizations, arXiv:1706.00393 [math.NT], 2017.

M. D. Schmidt, New recurrence relations and matrix equations for arithmetic functions generated by Lambert series, arXiv:1701.06257 [math.NT], Acta Arith. (2017).

Formula

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.

Example

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,
  ...

Mathematica

(* 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

Prog

(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

Crossrefs

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

Sequence in context: A285328 A321030 A373514 * A266349 A219094 A362824

Adjacent sequences: A290526 A290527 A290528 * A290530 A290531 A290532

Keyword

nonn,tabl

Author

Maxie D. Schmidt, Aug 04 2017

Status

approved