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 A002175 Excess of number of divisors of 12n+1 of form 4k+1 over those of form 4k+3. (Formerly M0416 N0159) 26
 1, 2, 3, 2, 1, 2, 2, 4, 2, 2, 1, 0, 4, 2, 3, 2, 2, 4, 0, 2, 2, 0, 4, 2, 3, 0, 2, 6, 2, 2, 1, 2, 0, 2, 2, 2, 2, 4, 2, 0, 4, 4, 4, 0, 1, 2, 0, 4, 2, 0, 2, 2, 5, 2, 0, 2, 2, 4, 4, 2, 0, 2, 4, 2, 2, 0, 4, 0, 0, 2, 3, 2, 4, 2, 0, 4, 0, 6, 2, 4, 1, 0, 4, 2, 2, 2, 2, 0, 0, 2, 0, 2, 8, 2, 2, 0, 2, 4, 0, 4, 2, 2, 3, 2, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). Number of ways to write n as an ordered sum of 2 generalized pentagonal numbers. - Ilya Gutkovskiy, Aug 14 2017 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS John Cerkan, Table of n, a(n) for n = 0..10000 J. W. L. Glaisher, On the square of Euler's series, Proc. London Math. Soc., 21 (1889), 182-194. Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of (phi(-x^3) / chi(-x))^2 in powers of x where phi(), chi() are Ramanujan theta functions. Expansion of q^(-1/12) * (eta(q^2) * eta(q^3)^2 / (eta(q) * eta(q^6)))^2 in powers of q. - Michael Somos, Sep 19 2005 Euler transform of period 6 sequence [ 2, 0, -2, 0, 2, -2, ...]. - Michael Somos, Sep 19 2005 G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A258279. - Michael Somos, May 25 2015 From Michael Somos, Jun 02 2012: (Start) a(n) = A008441(3*n) = A121363(3*n) = A122865(4*n) = A122856(8*n). a(n) = A116604(6*n) = A125079(6*n) = A129447(6*n) = A138741(6*n). a(n) = A(12*n+1) where A = A002654, A008442, A035154, A035181, A035184, A112301, A113406, A113652, A121450, A122864, A125061, A129448, A132004, A134013, A134015, A138746, A138950, A138952, A163746.  (End) Contribution from Michael Somos, May 25 2015: (Start) a(n) = A258277(4*n) = A258278(8*n) = A258291(3*n). a(n) = - A258210(12*n + 1) = A258228(12*n + 1) = A258256(12*n + 1). 2*a(n) = A258279(12*n + 1) = - A258292(12*n + 1). (End) G.f.: (Sum_{k=-inf..inf} x^(k*(3*k-1)/2))^2. - Ilya Gutkovskiy, Aug 14 2017 EXAMPLE G.f. = 1 + 2*x + 3*x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + 4*x^7 + 2*x^8 + 2*x^9 + ... G.f. = q + 2*q^13 + 3*q^25 + 2*q^37 + q^49 + 2*q^61 + 2*q^73 + 4*q^85 + 2*q^97 + ... MATHEMATICA ed[n_]:=Module[{divs=Divisors[12n+1]}, Count[divs, _?(Mod[#, 4] == 1&)]- Count[divs, _?(Mod[#, 4]==3&)]]; Array[ed, 110, 0] (* Harvey P. Dale, Jul 01 2012 *) a[ n_] := If[ n < 0, 0, With[ {m = 12 n + 1}, Sum[ KroneckerSymbol[ 4, d], {d, Divisors[m]}]]]; (* Michael Somos, Apr 23 2014 *) a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2] QPochhammer[ x^3]^2 / (QPochhammer[ x] QPochhammer[ x^6]))^2, {x, 0, n}]; (* Michael Somos, Apr 23 2014 *) a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^3] / QPochhammer[ x, x^2])^2, {x, 0, n}]; (* Michael Somos, May 25 2015 *) PROG (PARI) {a(n) = if( n<0, 0, n = 12*n + 1; sumdiv( n, d, (d%4==1) - (d%4==3)))}; /* Michael Somos, Sep 19 2005 */ (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A)^2 / (eta(x + A) * eta(x^6 + A)))^2, n))}; /* Michael Somos, Jun 02 2012 */ CROSSREFS Cf. A002654, A008441, A008442, A035154, A035181, A035184, A112301, A113406. Cf. A113652, A116604, A121363, A121450, A122856, A122864, A122865, A125061. Cf. A125079, A129447, A129448, A132004, A134013, A134015, A138741, A138746. Cf. A138950, A138952, A163746, A258210, A258228, A258256, A258279, A258292. Sequence in context: A265847 A260342 A281939 * A170823 A068073 A324504 Adjacent sequences:  A002172 A002173 A002174 * A002176 A002177 A002178 KEYWORD nonn AUTHOR STATUS approved

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Last modified June 16 21:20 EDT 2019. Contains 324155 sequences. (Running on oeis4.)