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A000025 Coefficients of the 3rd order mock theta function f(q).
(Formerly M0433 N0164)
18

%I M0433 N0164

%S 1,1,-2,3,-3,3,-5,7,-6,6,-10,12,-11,13,-17,20,-21,21,-27,34,-33,36,

%T -46,51,-53,58,-68,78,-82,89,-104,118,-123,131,-154,171,-179,197,-221,

%U 245,-262,279,-314,349,-369,398,-446,486,-515,557,-614,671,-715,767,-845,920,-977,1046,-1148,1244

%N Coefficients of the 3rd order mock theta function f(q).

%C a(n) = number of partitions of n with even rank minus number with odd rank. The rank of a partition is its largest part minus the number of parts.

%D G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 82, Examples 4 and 5.

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

%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 17, 31.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%H T. D. Noe, <a href="/A000025/b000025.txt">Table of n, a(n) for n = 0..1000</a>

%H G. E. Andrews, <a href="http://www.jstor.org/stable/2331943">An introduction to Ramanujan's "lost" notebook</a>, Amer. Math. Monthly 86 (1979), no. 2, 89-108. See page 95

%H L. A. Dragonette, <a href="http://dx.doi.org/10.1090/S0002-9947-1952-0049927-8">Some asymptotic formulas for the Mock Theta Series of Ramanujan</a>, Trans. Amer. Math. Soc., 72 (1952), 474-500.

%H John F. R. Duncan, Michael J. Griffin and Ken Ono, <a href="http://arxiv.org/abs/1503.01472">Proof of the Umbral Moonshine Conjecture</a>, arXiv:1503.01472, 2015. [See f(q)]

%H N. J. Fine, <a href="http://www.ams.org/bookstore?fn=20&amp;arg1=survseries&amp;ikey=SURV-27">Basic Hypergeometric Series and Applications</a>, Amer. Math. Soc., 1988; p. 55, Eq. (26.11), (26.24).

%H K. Ono, <a href="http://www.ams.org/notices/201011/rtx101101410p.pdf">The last words of a genius</a>, Notices Amer. math. Soc., 57 (2010), 1410-1419.

%H George N. Watson, <a href="http://jlms.oxfordjournals.org/content/s1-11/1/55.extract">The final problem: an account of the mock theta functions</a>, J. London Math. Soc., 11 (1936) 55-80

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MockThetaFunction.html">Mock Theta Function</a>

%F G.f.: 1 + Sum_{n>0} (q^(n^2) / Product_{i=1..n} (1 + q^i)^2).

%F G.f.: (1 + 4 * Sum_{n>0} (-1)^n * q^(n*(3*n+1)/2) / (1 + q^n)) / Product_{i>0} (1 - q^i).

%e G.f. = 1 + q - 2*q^2 + 3*q^3 - 3*q^4 + 3*q^5 - 5*q^6 + 7*q^7 - 6*q^8 + 6*q^9 + ...

%p series(1+4*add( (-1)^n*q^(n*(3*n+1)/2)/(1+q^n), n=1..71),q,71)/series(mul(1-q^i,i=1..71),q,71);

%t CoefficientList[Series[(1+4Sum[(-1)^n q^(n(3n+1)/2)/(1+q^n), {n, 1, 10}])/Sum[(-1)^n q^(n(3n+1)/2), {n, -8, 8}], {q, 0, 100}], q] (* _N. J. A. Sloane_ *)

%t sgn[P_ (* a partition *)] :=

%t Signature[

%t PermutationList[

%t Cycles[Flatten[

%t SplitBy[Range[Total[P]], (Function[{x}, x > #1] &) /@

%t Accumulate[P]], Length[P] - 1]]]]

%t conjugate[P_List(* a partition *)] :=

%t Module[{s = Select[P, #1 > 0 &], i, row, r}, row = Length[s];

%t Table[r = row; While[s[[row]] <= i, row--]; r, {i, First[s]}]]

%t Total[Function[{x}, sgn[x] sgn[conjugate[x]]] /@

%t IntegerPartitions[#]] & /@ Range[20]

%t (* _George Beck_, Oct 25 2014 *)

%t a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^k^2 / Product[ 1 + x^j, {j, k}]^2, {k, 0, Sqrt@n}], {x, 0, n}]]; (* _Michael Somos_, Jun 30 2015 *)

%o (PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=1, sqrtint(n), x^k^2 / prod(i=1, k, 1 + x^i, 1 + x * O(x^(n - k^2)))^2, 1), n))}; /* _Michael Somos_, Sep 02 2007 */

%Y Other '3rd order' mock theta functions are at A013953, A053250, A053251, A053252, A053253, A053254, A053255. See also A000039, A000199.

%K sign,easy,nice

%O 0,3

%A _N. J. A. Sloane_

%E Entry improved by comments from _Dean Hickerson_

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Last modified March 19 18:49 EDT 2019. Contains 321330 sequences. (Running on oeis4.)