A259551 Expansion of f(x^2, x^3) * f(-x^4, -x^6) / f(-x^2) in powers of x where f(,) is the Ramanujan general theta function.

1, 0, 2, 1, 2, 1, 2, 1, 3, 2, 4, 4, 5, 4, 6, 5, 7, 6, 9, 8, 11, 11, 13, 13, 17, 15, 20, 19, 23, 23, 27, 27, 33, 33, 38, 39, 45, 45, 53, 54, 62, 63, 73, 74, 84, 86, 97, 100, 112, 115, 130, 134, 148, 154, 170, 176, 195, 202, 222, 232, 255, 264, 290, 301, 329

Offset

0,3

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, 9th equation.

Links

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

G. E. Andrews, An introduction to Ramanujan's "lost" notebook, Amer. Math. Monthly 86 (1979), no. 2, 89-108. See page 97, Equation (3.5)

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of f(x^2, x^3) * G(x^2) in powers of x where f(,) is the Ramanujan general theta function and G() is a Rogers-Ramanujan function.

Euler transform of period 10 sequence [ 0, 2, 1, -1, -1, -1, 1, 2, 0, -1, ...].

G.f.: Product_{k>0} (1 - x^(5*k)) * (1 + x^(5*k - 3)) * (1 + x^(5*k - 2)) / ((1 - x^(10*k - 8)) * (1 - x^(10*k - 2))).

G.f.: (Sum_{k in Z} x^(k*(5*k + 1)/2)) / (Product_{k in Z} 1 - x^abs(10*k + 2)). - Michael Somos, Jul 09 2015

(-1)^n * A053258(n) + A053266(n) = a(n) unless n=0. Michael Somos, Jul 09 2015

A259910(n) = 2*A255065(n) + a(n). Michael Somos, Jul 09 2015

Example

G.f. = 1 + 2*x^2 + x^3 + 2*x^4 + x^5 + 2*x^6 + x^7 + 3*x^8 + 2*x^9 + ...
G.f. = 1/q + 2*q^239 + q^359 + 2*q^479 + q^599 + 2*q^719 + q^839 + ...

Mathematica

a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ 0, -2, -1, 1, 1, 1, -1, -2, 0, 1}[[Mod[k, 10, 1]]], {k, n}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[x^5] QPochhammer[ -x^2, x^5] QPochhammer[ -x^3, x^5] / (QPochhammer[ x^2, x^10]  QPochhammer[ x^8, x^10]), {x, 0, n}];

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 1, 0, -2, -1, 1, 1, 1, -1, -2, 0][k%10 + 1]), n))};

Crossrefs

Cf. A053258, A053266, A255065, A259910.

Sequence in context: A161259 A161284 A185215 * A058232 A216320 A308455

Adjacent sequences: A259548 A259549 A259550 * A259552 A259553 A259554

Keyword

nonn

Author

Michael Somos, Jun 30 2015

Status

approved