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 A259920 Expansion of phi(-x^5) * f(-x^5) / f(-x, -x^4) in powers of x where phi() and f() are Ramanujan theta functions. 1
 1, 1, 1, 1, 2, 0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 2, 3, 1, 3, 2, 5, 3, 5, 2, 6, 3, 6, 3, 7, 4, 7, 5, 9, 5, 9, 5, 11, 6, 11, 7, 14, 7, 15, 9, 17, 9, 17, 9, 21, 11, 21, 12, 25, 13, 25, 15, 29, 16, 31, 17, 35, 19, 37, 21, 42, 22, 44, 25, 49, 27, 52, 29, 58, 32, 61 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106). REFERENCES Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, 8th equation. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of f(-x^5)^3 / (f(-x^10) * f(-x^2, -x^3)) in powers of x where f(,) is the Ramanujan general theta function. Expansion of phi(-x^5) * G(x) in powers of x where f(,) is the Ramanujan general theta function and G() is a Rogers-Ramanujan function. - Michael Somos, Jul 09 2015 Euler transform of period 10 sequence [ 1, 0, 0, 1, -2, 1, 0, 0, 1, -1, ...]. G.f.: (Sum_{k in Z} (-1)^k * x^(5*k^2)) / (Product_{k in Z} 1 - x^abs(5*k + 1)). EXAMPLE G.f. = 1 + x + x^2 + x^3 + 2*x^4 + x^6 + x^7 + 2*x^8 + x^9 + 2*x^10 + x^11 + ... G.f. = q^-1 + q^59 + q^119 + q^179 + 2*q^239 + q^359 + q^419 + 2*q^479 + q^539 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^5] / (QPochhammer[ x, x^5] QPochhammer[ x^4, x^5]), {x, 0, n}]; a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ -1, 0, 0, -1, 2, -1, 0, 0, -1, 1}[[Mod[k, 10, 1]]], {k, n}], {x, 0, n}]; PROG (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^ [1, -1, 0, 0, -1, 2, -1, 0, 0, -1][k%10+1]), n))}; CROSSREFS Cf. A053256, A053266. Sequence in context: A066360 A061358 A025866 * A048881 A026931 A127506 Adjacent sequences:  A259917 A259918 A259919 * A259921 A259922 A259923 KEYWORD nonn AUTHOR Michael Somos, Jul 08 2015 STATUS approved

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Last modified February 25 06:46 EST 2020. Contains 332220 sequences. (Running on oeis4.)