A000039 Coefficient of q^(2n) in the series expansion of Ramanujan's mock theta function f(q).

1, -2, -3, -5, -6, -10, -11, -17, -21, -27, -33, -46, -53, -68, -82, -104, -123, -154, -179, -221, -262, -314, -369, -446, -515, -614, -715, -845, -977, -1148, -1321, -1544, -1778, -2060, -2361, -2736, -3121, -3592, -4097, -4696, -5340, -6105, -6916, -7882, -8919, -10123, -11429, -12952, -14580

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms 0..1000 from T. D. Noe)
• L. A. Dragonette, Some Asymptotic Formulae for the Mock Theta Series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952), 474-500.
• Eric Weisstein's World of Mathematics, Mock Theta Function

Crossrefs

A000025(2n)=a(n). Cf. A000199.

Programs

Mathematica

f[q_, s_] := Sum[q^(n^2)/Product[1+q^k, {k, n}]^2, {n, 0, s}]; Take[CoefficientList[Series[f[q, 100], {q, 0, 100}], q], {1, -1, 2}]

PARI

a(n)=if(n<0,0,polcoeff(1+sum(k=1,sqrtint(2*n),x^k^2/prod(i=1,k,1+x^i,1+O(x^(2*n)))^2),2*n))

Formula

a(n) ~ -exp(Pi*sqrt(n/3)) / (2*sqrt(2*n)). - Vaclav Kotesovec, Jun 12 2019

Extensions

More terms from Eric W. Weisstein