A064053 Auxiliary sequence gamma(n) used to compute coefficients in series expansion of the mock theta function f(q) via A(n) = Sum_{r=0..n} p(r)*gamma(n-r), with p(r) the partition function A000041.

1, 0, -4, 4, -4, 4, -4, 8, -4, 0, -4, 8, -4, 0, -4, 4, -4, 0, 0, 8, -4, -4, -4, 8, 0, 0, 0, 4, -4, 0, -4, 8, -4, -4, 0, 8, 0, 0, -8, 4, -8, 0, 4, 8, -4, 0, -8, 8, 0, 0, -4, 4, -4, 0, -4, 12, -4, 0, 0, 8, -4, 0, -8, 0, -4, 4, 4, 8, -4, 0, -12, 8, 0, 0, 0, 4, -4, -4, -4, 8, -8, 0, 0, 8, 4, 4, -8, 0, -4, 0, 0, 4, -4, 0, -8, 12, 0, 0, 4, 0, -4, 0, -4

Offset: 0

Author

Eric W. Weisstein, Aug 28 2001

Keywords

• sign

Comments

See Dragonette for the definition of f(q) and A(n). - N. J. A. Sloane, Sep 24 2022

Examples

			G.f. = 1 - 4*x^2 + 4*x^3 - 4*x^4 + 4*x^5 - 4*x^6 + 8*x^7 - 4*x^8 - 4*x^10 + 8*x^11 - 4*x^12 - ...
		

References

• G. E. Andrews, The theory of partitions, Cambridge University Press, Cambridge, 1998, page 82, Example 5. MR1634067 (99c:11126). [The Gamma function used by Andrews is the classical Gamma function, which is different from the gamma(n) of this sequence. - N. J. A. Sloane, Sep 24 2022]

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• L. A. Dragonette, Some Asymptotic Formulae for the Mock Theta Series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952), 474-500. See page 496.
• L. A. Dragonette, Some Asymptotic Formulae for the Mock Theta Series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952), 474-500. Enlargement of a portion of page 496 in order to show correct spelling of gamma(n).
• Eric Weisstein's World of Mathematics, Mock Theta Function.

Crossrefs

Cf. A000025, A000039, A000041, A000199, A096661.

Programs

Mathematica

a[ n_]:= SeriesCoefficient[1 +4 *Sum[(-1)^k*x^(k*(3*k+1)/2)/(1+x^k), {k, Quotient[Sqrt[1 +24*n] - 1, 6]}], {x, 0, n}]; (* Michael Somos, Apr 08 2015 *)

PARI

{a(n) = if( n<1, n==0, 4 * polcoeff( sum(k=1, (sqrtint(24*n + 1) - 1) \ 6, (-1)^k * x^((3*k^2 + k)/2) / (1 + x^k), x * O(x^n)), n))}; /* Michael Somos, Mar 13 2006 */

Formula

G.f.: 1 + 4 * Sum_{k>0} (-1)^k * x^(k*(3*k + 1)/2) / (1 + x^k). - Michael Somos, Jun 19 2003

Convolution of this sequence and A000041 is A000025. - Michael Somos, Jun 19 2003

a(n) = 4 * A096661(n) unless n=0.

Extensions

Entry revised by Michael Somos, Mar 13 2006

Deleted edit that tried to change gamma(n) to Gamma(n), and restored original definition. - N. J. A. Sloane, Sep 24 2022