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A064053 Dragonette's sequence gamma(n). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Gamma(n) is used to compute coefficients in series expansion of mock theta function f(q) via A[n]==Sum[P[r]gamma[n-r],{r,0,n}], with P the partition function A000041.

REFERENCES

G. E. Andrews, The theory of partitions, Cambridge University Press, Cambridge, 1998, page 82, Example 5. MR1634067 (99c:11126)

L. A. Dragonette, Some asymptotic formulas for the Mock Theta Series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952), 474-500. see page 496

LINKS

Table of n, a(n) for n=0..102.

Eric Weisstein's World of Mathematics, Mock Theta Function.

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.

EXAMPLE

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 - ...

MATHEMATICA

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

PROG

(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 */

CROSSREFS

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

Sequence in context: A103276 A059190 A085142 * A108893 A162281 A262690

Adjacent sequences:  A064050 A064051 A064052 * A064054 A064055 A064056

KEYWORD

sign

AUTHOR

Eric W. Weisstein, Aug 28 2001

EXTENSIONS

Entry revised by Michael Somos, Mar 13 2006

STATUS

approved

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Last modified December 5 05:27 EST 2016. Contains 278761 sequences.