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

1,9

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.

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 20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

Formula

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

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

Example

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 + ...

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 *)

Prog

(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))};

Crossrefs

Cf. A053265, A053267.

Sequence in context: A212813 A112219 A035458 * A357457 A194902 A194874

Adjacent sequences: A259768 A259769 A259770 * A259772 A259773 A259774

Keyword

nonn

Author

Michael Somos, Jul 04 2015

Status

approved