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

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

Author

Maxie D. Schmidt, Aug 04 2017

Keywords

• sign

Comments

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.

Crossrefs

Cf. A051424.

Programs

Mathematica

(* 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]

Formula

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.