A260971 Expansion of phi_0(-q) in powers of q where phi_0() is a 5th-order mock theta function.

1, -1, 1, 0, 1, -1, 0, -1, 1, -1, 1, 0, 1, -1, 1, -1, 1, -2, 1, -1, 1, -1, 1, -1, 2, -2, 2, -1, 2, -2, 1, -2, 2, -2, 2, -2, 2, -3, 2, -2, 3, -3, 3, -2, 3, -3, 3, -3, 3, -4, 4, -3, 4, -4, 3, -4, 4, -5, 4, -4, 5, -5, 5, -5, 6, -6, 5, -5, 6, -6, 6, -6, 7, -7, 7

Offset

0,18

Links

G. C. Greubel, Table of n, a(n) for n = 0..2500

Formula

G.f.: Sum_{k >= 0} (-x)^n^2 * (1 - x) * (1 - x^3) * ... * (1 - x^(2*k-1)).

a(n) = (-1)^n * A053258(n) = 2 * A053264(n) - A053262(n).

a(n) ~ (-1)^n * sqrt(phi) * exp(Pi*sqrt(n/30)) / (2*5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 15 2019

Example

G.f. = 1 - x + x^2 + x^4 - x^5 - x^7 + x^8 - x^9 + x^10 + x^12 - x^13 + ...
G.f. = q^-1 - q^119 + q^239 + q^479 - q^599 - q^839 + q^959 - q^1079 + ...

Mathematica

a[n_]:= SeriesCoefficient[Sum[(-x)^( k^2)*Product[1 - x^(2*j - 1), {j, 1, k}], {k, 0, Sqrt[n]}], {x, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Aug 01 2018 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, sqrtint(n), (-x)^k^2 * prod(i=1, k, 1 - x^(2*i - 1), 1 + x * O(x^(n - k^2)))), n))};

Crossrefs

Cf. A053258, A053262, A053264.

Sequence in context: A031262 A047072 A178058 * A053258 A350738 A053632

Adjacent sequences: A260968 A260969 A260970 * A260972 A260973 A260974

Keyword

sign

Author

Michael Somos, Aug 06 2015

Status

approved