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A053256 Coefficients of the '5th order' mock theta function f_0(q). 13
1, 1, -1, 1, 0, 0, -1, 1, 0, 1, -2, 1, -1, 2, -2, 2, -1, 1, -3, 2, -1, 3, -3, 2, -2, 3, -4, 3, -3, 4, -5, 5, -3, 5, -7, 5, -5, 6, -7, 7, -6, 7, -9, 9, -7, 9, -11, 9, -9, 11, -13, 12, -11, 13, -15, 15, -13, 16, -19, 17, -17, 19, -21, 21, -20, 22, -26, 25, -23, 27, -30, 29, -28, 32, -35, 34, -34, 36, -41, 40, -38, 44, -48, 46 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,11

COMMENTS

In Ramanujan's lost notebook page 21 is written the g.f. neatly crossed out between the 3rd and 4th equations. - Michael Somos, Feb 13 2017

REFERENCES

George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134

George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255

Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 21, 22, 23

George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304

Dean Hickerson, A proof of the mock theta conjectures, Inventiones Mathematicae, 94 (1988) 639-660

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000

FORMULA

G.f.: 1 + Sum_{k>0} q^k^2 / ((1 + q) * (1 + q^2) * ... * (1 + q^k)).

Consider partitions of n into parts differing by at least 2. For n>0: a(n) = number of them with largest part odd minus number with largest part even.

EXAMPLE

G.f. = 1 + x - x^2 + x^3 - x^6 + x^7 + x^9 - 2*x^10 + x^11 - x^12 + 2*x^13 - ...

MATHEMATICA

Series[Sum[q^n^2/Product[1+q^k, {k, 1, n}], {n, 0, 10}], {q, 0, 100}]

a[ n_] := SeriesCoefficient[ Sum[ x^k^2 / QPochhammer[ -x, x, k] // FunctionExpand, {k, 0, Sqrt@ n}], {x, 0, n}]; (* Michael Somos, Feb 13 2017 *)

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 + O(x^(n - (k-1)^2 + 1))), 1), n))}; /* Michael Somos, Mar 12 2006 */

CROSSREFS

Other '5th order' mock theta functions are at A053257, A053258, A053259, A053260, A053261, A053262, A053263, A053264, A053265, A053266, A053267.

Sequence in context: A262611 A110535 A033941 * A102418 A106032 A003646

Adjacent sequences:  A053253 A053254 A053255 * A053257 A053258 A053259

KEYWORD

sign,easy

AUTHOR

Dean Hickerson, Dec 19 1999

STATUS

approved

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Last modified May 27 00:21 EDT 2017. Contains 287189 sequences.