A259771 Expansion of x * psi(x^5) * f(-x^10) / f(-x^2,-x^8) in powers of x where psi(), f() are Ramanujan theta functions.

1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 3, 4, 4, 5, 4, 6, 5, 7, 7, 9, 8, 10, 10, 12, 12, 15, 14, 18, 17, 20, 20, 24, 24, 28, 28, 33, 33, 38, 38, 44, 45, 50, 52, 59, 60, 68, 69, 78, 80, 89, 92, 102, 105, 116, 120, 133, 137, 151, 156, 171, 178, 194, 201

Offset: 1

Author

Michael Somos, Jul 04 2015

Keywords

• nonn

Comments

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

The g.f. for this sequence is the last term of the 14th equation on page 20 of Ramanujan 1988.

Examples

			G.f. = x + x^3 + x^5 + x^6 + x^7 + x^8 + 2*x^9 + x^10 + 2*x^11 + x^12 + ...
G.f. = q^49 + q^289 + q^529 + q^649 + q^769 + q^889 + 2*q^1009 + q^1129 + ...
		

References

• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 20

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A053265, A053267.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ x Product[ (1 - x^k)^{ 0, -1, 0, 0, -1, 0, 0, -1, 0, 1}[[Mod[k, 10, 1]]], {k, n}], {x, 0, n}];
QP:= QPochhammer; a[n_]:= SeriesCoefficient[ x*QP[x^10]/(QP[x^5, x^10]* QP[x^2, x^10]*QP[x^8, x^10]), {x, 0, n}]; Table[a[n], {n, 1, 100}] (* G. C. Greubel, Mar 16 2018 *)

PARI

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

Formula

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

a(n) = A053265(n-1) - A053267(n).