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A053266 Coefficients of the '5th order' mock theta function Phi(q) 15
0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 8, 8, 9, 10, 12, 12, 14, 15, 17, 18, 20, 21, 25, 26, 29, 31, 35, 36, 41, 43, 48, 51, 56, 59, 66, 70, 76, 81, 89, 94, 103, 109, 119, 126, 137, 144, 158, 167, 180, 191, 207, 218, 236, 250, 269, 285, 306, 323, 349, 368 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

In Ramanujan's lost notebook the generating function is denoted by phi(q) on pages 18 and 20, however on page 18 there is no minus one first term. - ~~~

REFERENCES

G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, MR2952081, See p. 12, Equation (2.1.18) and also page 26 equation (2.4.8).

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

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

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 18, 20, 23

LINKS

Table of n, a(n) for n=0..66.

FORMULA

G.f.: -1 + Sum_{k>=0} q^(5k^2)/((1-q)(1-q^4)(1-q^6)(1-q^9)...(1-q^(5k+1))).

3*a(n) = A053262(n) + A259910(n) unless n=0. - Michael Somos, Jul 07 2015

EXAMPLE

G.f. = x + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 2*x^8 + 3*x^9 + ...

MATHEMATICA

Series[Sum[q^(5n^2)/Product[1-q^Abs[5k+1], {k, -n, n}], {n, 0, 4}], {q, 0, 100}]-1

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ -1 + Sum[ x^(5 k^2) / (QPochhammer[ x, x^5, k + 1] QPochhammer[ x^4, x^5, k]), {k, 0, Sqrt[n/5]}], {x, 0, n}]]; (* Michael Somos, Jul 07 2015 *)

a[ n_] := If[ n < 0, 0, With[ {m = Sqrt[1 + 24 n/5]}, SeriesCoefficient[ -1 + Sum[ (-1)^k x^(5 k (3 k + 1)/2) / (1 - x^(5 k + 1)), {k, Quotient[m + 1, -6], Quotient[m - 1, 6]}] / QPochhammer[ x^5], {x, 0, n}]]]; (* Michael Somos, Jul 07 2015 *)

PROG

(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, sqrtint(n\5), x^(5*k^2) / prod(i=1, 5*k+1, 1 - if( i%5==1 || i%5==4, x^i), 1 + x * O(x^(n - 5*k^2)))) - 1, n))}; /* Michael Somos, Jul 07 2015 */

(PARI) {a(n) = my(A, m); if( n<0, 0, m = sqrtint(1 + 24*n\5); A = x * O(x^n); polcoeff( sum(k=(m + 1)\-6, (m - 1)\6, (-1)^k * x^(5*k*(3*k + 1)/2) / (1 - x^(5*k + 1)), A) / eta(x^5 + A) - 1, n))}; /* Michael Somos, Jul 07 2015 */

CROSSREFS

Other '5th order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053261, A053262, A053263, A053264, A053265, A053267.

Cf. A259910.

Sequence in context: A185322 A068980 A279135 * A112217 A172033 A214131

Adjacent sequences:  A053263 A053264 A053265 * A053267 A053268 A053269

KEYWORD

nonn,easy

AUTHOR

Dean Hickerson, Dec 19 1999

STATUS

approved

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Last modified December 7 17:16 EST 2016. Contains 278890 sequences.