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 A290529 Inverse of the factorization matrix in the Lambert series factorization theorem. 0
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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: A294895 A285328 A321030 * A266349 A219094 A213268 Adjacent sequences:  A290526 A290527 A290528 * A290530 A290531 A290532 KEYWORD nonn,tabl AUTHOR Maxie D. Schmidt, Aug 04 2017 STATUS approved

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Last modified September 27 08:39 EDT 2021. Contains 347689 sequences. (Running on oeis4.)