A260460 Expansion of f(-q) in powers of q where f() is a 3rd order mock theta function.

1, -1, -2, -3, -3, -3, -5, -7, -6, -6, -10, -12, -11, -13, -17, -20, -21, -21, -27, -34, -33, -36, -46, -51, -53, -58, -68, -78, -82, -89, -104, -118, -123, -131, -154, -171, -179, -197, -221, -245, -262, -279, -314, -349, -369, -398, -446, -486, -515, -557

Offset

0,3

Links

Table of n, a(n) for n=0..49.

Formula

G.f.: Sum_{k>=0} (-x)^(k^2) / Product_{i=1..k} (1 + (-x)^i)^2.

G.f.: 2 * (Sum_{k in Z} (-1)^k * x^(k*(3*k + 1)/2) / (1 + x^k)) / (Sum_{k in Z} (-1)^k * x^(k*(3*k + 1)/2))

a(n) = (-1)^n * A000025(n). a(n) < 0 if n>0.

a(n) = A053250(n) - 2 * A053251(n) = 2 * A053250(n) - A132969(n) = A132969(n) - 4 * A053251(n).

Example

G.f. = 1 - x - 2*x^2 - 3*x^3 - 3*x^4 - 3*x^5 - 5*x^6 - 7*x^7 - 6*x^8 + ...

Mathematica

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ (-x)^k^2 / Product[ 1 + (-x)^j, {j, k}]^2, {k, 0, Sqrt@n}], {x, 0, n}]];

Prog

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

Crossrefs

Cf. A000025, A053250, A053251, A132969.

Sequence in context: A263253 A162157 A060210 * A000025 A036020 A036024

Adjacent sequences: A260457 A260458 A260459 * A260461 A260462 A260463

Keyword

sign

Author

Michael Somos, Jul 26 2015

Status

approved