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 A113429 Expansion of f(-x, -x^4) in powers of x where f(, ) is Ramanujan's general theta function. 7
 1, -1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS For the g.f. identity see the Hardy-Wright reference, Theorem 355 on p. 284. - Wolfdieter Lang, Oct 28 2016 REFERENCES G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 93. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, p. 284. LINKS Seiichi Manyama, Table of n, a(n) for n = 0..10000 Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Euler transform of period 5 sequence [-1, 0, 0, -1, -1, ...]. |a(n)| is the characteristic function of A085787. G.f.: Product_{k>0} (1 - x^(5*k)) * (1 - x^(5*k-1)) * (1 - x^(5*k-4)) = Sum_{k in Z} (-1)^k * x^((5*k^2+3*k)/2). f(a, b) = Sum_{k in Z} a^((k^2+k)/2) * b^((k^2-k)/2) is Ramanujan's general theta function. G.f.: Sum_{n>=0} (x^(n*(n+1)) * Product_{k>=n+1} (1-x^k)). - Joerg Arndt, Apr 07 2011 From Wolfdieter Lang, Oct 30 2016: (Start) a(n) = (-1)^k if n = b(2*k) for k >= 0, a(n) = (-1)^k if n = b(2*k-1), for k >= 1, and a(n) = 0 otherwise, where b(n) = A085787(n). See the second formula. G.f.: Sum_{n>=0} (-1)^n*x^(n*(5*n+3)/2)*(1-x^(2*n+1)). See the Hardy reference, p. 93, G_1(x,x) from eq. (6.11.1) with C_n(x,x) = 1. (End) G.f.: Sum_{n>=0} (-1)^n*x^(n*(5*n-3)/2)*(1-x^(4*(2*n+1)). Reordered G_1(x,x) from the preceding formula. This is G_4(x,x) from Hardy, p. 93, eq. (6.11.1) with C_n(x,x) = 1. Note that Hardy uses only G_0, G_1 and G_2. - Wolfdieter Lang, Nov 01 2016 a(n) = -(1/n)*Sum_{k=1..n} A284361(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017 EXAMPLE G.f. = 1 - x - x^4 + x^7 + x^13 - x^18 - x^27 + x^34 + x^46 - x^55 - x^70 + ... G.f. = q^9 - q^49 - q^169 + q^289 + q^529 - q^729 - q^1089 + q^1369 + q^1849 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^5] QPochhammer[ x^4, x^5] QPochhammer[ x^5], {x, 0, n}]; (* Michael Somos, Jun 26 2017 *) a[ n_] := Module[{m = 40 n + 9, k}, If[IntegerQ[k = Sqrt[m]], If[Mod[k, 10] == 7, k = -k]; (-1)^Quotient[k, 10], 0]]; (* Michael Somos, Jun 26 2017 *) PROG (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 - x^k*[1, 1, 0, 0, 1][k%5 + 1], 1 + x * O(x^n)), n))}; (PARI) {a(n) = my(m, k); if( n<0, 0, issquare(m = 40*n + 9, &k), if( k%10==7, k=-k); (-1)^(k\10), 0)}; /* Michael Somos, Oct 29 2016 */ CROSSREFS Cf. A085787. Sequence in context: A087049 A186447 A118009 * A133100 A216230 A077606 Adjacent sequences:  A113426 A113427 A113428 * A113430 A113431 A113432 KEYWORD sign,easy AUTHOR Michael Somos, Oct 31 2005 STATUS approved

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Last modified January 21 19:08 EST 2019. Contains 319350 sequences. (Running on oeis4.)