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A000025 Coefficients of the 3rd order mock theta function f(q).
(Formerly M0433 N0164)
13
1, 1, -2, 3, -3, 3, -5, 7, -6, 6, -10, 12, -11, 13, -17, 20, -21, 21, -27, 34, -33, 36, -46, 51, -53, 58, -68, 78, -82, 89, -104, 118, -123, 131, -154, 171, -179, 197, -221, 245, -262, 279, -314, 349, -369, 398, -446, 486, -515, 557, -614, 671, -715, 767, -845, 920, -977, 1046, -1148, 1244 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.

Let p(n) be the number of partitions of n. There are p(n) conjugacy classes in the symmetric group S_n. The lengths of the cycles of a cycle decomposition of any permutation in the class C corresponding to a partition P are given by the parts of P. Therefore the sign (+1 or -1) of the permutations in C is unique; define that to be the sign of the partition, sgn(P). Let P' be the conjugate of P. Then it seems that a(n) is the sum over all partitions P of n of sgn(P)*sgn(P'). - George Beck, Oct 25 2014

REFERENCES

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

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 17, 31.

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

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

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

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

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 55, Eq. (26.11), (26.24).

K. Ono, The last words of a genius, Notices Amer. math. Soc., 57 (2010), 1410-1419.

George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80

Eric Weisstein's World of Mathematics, Mock Theta Function

FORMULA

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

EXAMPLE

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 + ...

MAPLE

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);

MATHEMATICA

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 *)

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

Signature[

  PermutationList[

   Cycles[Flatten[

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

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

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

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

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

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

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

(* George Beck, Oct 25 2014 *)

PROG

(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 */

CROSSREFS

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

Sequence in context: A029065 A162157 A060210 * A036020 A036024 A036029

Adjacent sequences:  A000022 A000023 A000024 * A000026 A000027 A000028

KEYWORD

sign,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Entry improved by comments from Dean Hickerson

STATUS

approved

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Last modified December 22 14:10 EST 2014. Contains 252364 sequences.