A255065 Expansion of x * psi(x^5) * f(-x^10) / f(-x^4, -x^6) in powers of x where psi(), f() are Ramanujan theta functions.

1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 4, 4, 4, 6, 5, 6, 6, 8, 8, 10, 9, 11, 11, 13, 13, 16, 15, 17, 18, 21, 21, 24, 24, 28, 29, 32, 33, 38, 38, 43, 44, 49, 51, 57, 58, 65, 67, 73, 76, 85, 87, 95, 99, 109, 113, 123, 127, 139, 145, 157

Offset

1,13

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, 7th equation.

Links

Table of n, a(n) for n=1..71.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of x * f(-x, -x^9) * f(-x^10) / f(-x, -x^4) in powers of x where f(,) is the Ramanujan general theta function.

Expansion of x * psi(x^5) * H(x^2) in powers of x where f(,) is the Ramanujan general theta function and H() is a Rogers-Ramanujan function. - Michael Somos, Jul 09 2015

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

G.f.: x * (Sum_{k>0} x^(5*k*(k-1)/2)) / (Product_{k in Z} 1 - x^abs(10*k + 4)).

A053266(n) = A053264(n) + a(n) unless n=0.

Example

G.f. = x + x^5 + x^6 + x^7 + x^9 + x^10 + x^11 + x^12 + 2*x^13 + x^14 + ...
G.f. = q^119 + q^599 + q^719 + q^839 + q^1079 + q^1199 + q^1319 + q^1439 + ...

Mathematica

a[ n_] := SeriesCoefficient[ Product[ x * (1 - x^k)^{ 0, 0, 0, -1, -1, -1, 0, 0, 0, 1} [[Mod[k, 10, 1]]], {k, n}], {x, 0, n}];

Prog

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

Crossrefs

Cf. A053264, A053266.

Sequence in context: A161090 A349219 A178697 * A027349 A053277 A078661

Adjacent sequences: A255062 A255063 A255064 * A255066 A255067 A255068

Keyword

nonn

Author

Michael Somos, Jul 08 2015

Status

approved