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A259774 Expansion of f(x, x^7) / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function. 2
1, 0, 0, -1, 1, -1, 1, -1, 2, -2, 2, -3, 4, -4, 4, -6, 7, -7, 8, -10, 12, -13, 14, -17, 21, -22, 24, -29, 33, -36, 40, -46, 53, -58, 63, -73, 83, -90, 99, -113, 127, -138, 152, -171, 191, -209, 228, -255, 285, -309, 338, -377, 416, -453, 495, -547, 603, -656 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

REFERENCES

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 15th equation.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of f(x^4, x^12) / f(x^3, x^5) where f(, ) is Ramanujan's general theta function.

Euler transform of period 16 sequence [ 0, 0, -1, 1, -1, 1, 0, 0, 0, 1, -1, 1, -1, 0, 0, 0, ...].

G.f.: (1 + x^4 + x^12 + x^24 + x^40 + ...) / (1 + x^3 + x^5 + x^14 + x^18 + ...). [Ramanujan]

G.f.: 1 - x^3 * (1 - x) / (1 - x^2) + x^8 * (1 - x) * (1 - x^3) / ((1 - x^2) * (1 - x^4)) - ... [Ramanujan]

a(n) = (-1)^n * A036015(n) = A029838(2*n + 1) = - A082303(2*n + 1).

Convolution product of A106507 and A214263.

EXAMPLE

G.f. = 1 - x^3 + x^4 - x^5 + x^6 - x^7 + 2*x^8 - 2*x^9 + 2*x^10 - 3*x^11 + ...

G.f. = q^7 - q^55 + q^71 - q^87 + q^103 - q^119 + 2*q^135 - 2*q^151 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^4, x^4] / (QPochhammer[ -x^3, x^8] QPochhammer[ -x^5, x^8]), {x, 0, n}];

a[ n_] := SeriesCoefficient[ 1 / Product[ (1 + x^(8 k + 3)) (1 - x^(8 k + 4)) (1 + x^(8 k + 5)), {k, 0, Ceiling[ n/8]}], {x, 0, n}];

a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ 0, 0, 1, -1, 1, -1, 0, 0, 0, -1, 1, -1, 1, 0, 0, 0}[[Mod[k, 16, 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))^[ 0, 0, 0, 1, -1, 1, -1, 0, 0, 0, -1, 1, -1, 1, 0, 0][k%16 + 1]), n))};

CROSSREFS

Cf. A029838, A036015, A082303, A106507, A214263.

Sequence in context: A084827 A251631 A029076 * A036015 A130521 A090619

Adjacent sequences:  A259771 A259772 A259773 * A259775 A259776 A259777

KEYWORD

sign

AUTHOR

Michael Somos, Nov 08 2015

STATUS

approved

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Last modified January 23 01:48 EST 2020. Contains 331166 sequences. (Running on oeis4.)