Maxie D. Schmidt - cp's OEIS frontend

User: Maxie D. Schmidt

Maxie D. Schmidt has authored 2 sequences.

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

Maxie D. Schmidt, Aug 04 2017

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.

			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,
  ...

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).

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

(* 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

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

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.

A286889 Sequence generated by the reciprocal of the generating function for A051424.

Original entry on oeis.org

1, -1, -1, 0, 1, 0, 1, -1, 0, 0, 1, -3, 2, 0, 3, -1, -2, -10, 8, 5, 8, -6, -3, -24, 17, 8, 12, -15, 19, -37, 18, -29, 18, 3, 109, -72, -28, -153, 46, 72, 335, -165, -86, -346, 84, -34, 650, -224, 245, -492, -69, -1054, 966, 161

Offset: 0

Maxie D. Schmidt, Aug 04 2017

sign

Inverts A051424 by discrete convolution: Sum_{k=0..n} rpp(k) rpp2(n-k) = delta_{n,0}. This is easy enough to see by the generating function definition of the sequence.

Cf. A051424.

(* For all the terms of the sequence A051424 listed in the database, the partial generating function for the sequence is given by:
  rpp2[n_] :=
  SeriesCoefficient[1/(1 + q + 2 q^2 + 3 q^3 + 4 q^4 + 6 q^5 + 7 q^6 + 10 q^7 +
     12 q^8 + 15 q^9 + 18 q^10 + 23 q^11 + 27 q^12 + 33 q^13 +
     38 q^14 + 43 q^15 + 51 q^16 + 60 q^17 + 70 q^18 + 81 q^19 +
     92 q^20 + 102 q^21 + 116 q^22 + 134 q^23 + 153 q^24 + 171 q^25 +
     191 q^26 + 211 q^27 + 236 q^28 + 266 q^29 + 301 q^30 +
     335 q^31 + 367 q^32 + 399 q^33 + 442 q^34 + 485 q^35 +
     542 q^36 + 598 q^37 + 649 q^38 + 704 q^39 + 771 q^40 +
     849 q^41 + 936 q^42 + 1023 q^43 + 1103 q^44 + 1185 q^45 +
     1282 q^46 + 1407 q^47 + 1535 q^48 + 1662 q^49 + 1790 q^50 +
     1917 q^51 + 2063 q^52 + 2245 q^53 + 2436 q^54), {q, 0, n}]
  Table[rpp2[n], {n, 0, 53}] *)
(* This generating function was created from the original sequence data by the following code: *)
  StringSplit["1, 1, 2, 3, 4, 6, 7, 10, 12, 15, 18, 23, 27, 33, 38, 43, 51, 60, 70, 81, 92, 102, 116, 134, 153, 171, 191, 211, 236, 266, 301, 335, 367, 399, 442,     485, 542, 598, 649, 704, 771, 849, 936, 1023, 1103, 1185, 1282, 1407, 1535, 1662, 1790, 1917, 2063, 2245, 2436", ", "]
  MapIndexed[ToExpression[(#1)] Power[q, First[#2] - 1] &, %]
  Apply[Plus, %]
  TeXForm@PolynomialForm[%, TraditionalOrder -> False]

Letting rpp(n) := A051424(n), and this sequence equal rpp2(n), we have the following two formulas for Euler's totient function:
phi(n) = Sum_{j=1..n} Sum_{k=1..j-1} Sum_{i=0..j-1-k} rpp_2(n-j) rpp(j-1-k-i) Iverson{(i+k+1, k)=1};
phi(n) = Sum_{d:(d,n)=1} (Sum_{k=1..d+1} Sum_{i=1..d} Sum_{j=2..k} rpp(k-j) rpp_2(i+1-k) mu_{d,i} phi(j)).
I prove that these expressions are correct in an article I have written which motivated the need for this sequence. A proof is available upon reasonable email request.

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User: Maxie D. Schmidt

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

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