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

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

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.

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 17, 31.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

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe and then corrected by Sean A. Irvine, Apr 25 2019)

G. E. Andrews, An introduction to Ramanujan's "lost" notebook, Amer. Math. Monthly 86 (1979), no. 2, 89-108. See page 95.

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

John F. R. Duncan, Michael J. Griffin and Ken Ono, Proof of the Umbral Moonshine Conjecture, arXiv:1503.01472 [math.RT], 2015. [See f(q)]

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>=1} (q^(n^2) / Product_{i=1..n} (1 + q^i)^2).

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

a(n) ~ -(-1)^n * exp(Pi*sqrt(n/6)) / (2*sqrt(n)) [Ramanujan]. - Vaclav Kotesovec, Jun 10 2019

G.f.: 1 - Sum_{n >= 1} (-1)^n*x^n/Product_{k = 1..n} 1 + x^k. See Fine, equation 26.22, p. 55. - Peter Bala, Feb 04 2021

From Seiichi Manyama, May 23 2023: (Start)

a(n) = A340601(n) - A340692(n).

G.f.: 1 + (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2) * (1-x^k)^2 / (1+x^k). (End)

Example

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 + ...

Maple

a:= m-> coeff(series((1+4*add((-1)^n*q^(n*(3*n+1)/2)/
        (1+q^n), n=1..m))/mul(1-q^i, i=1..m), q, m+1), q, m):
seq(a(n), n=0..120);

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 *)
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 *)

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 */
(PARI) my(N=60, x='x+O('x^N)); Vec(1+1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2)*(1-x^k)^2/(1+x^k))) \\ Seiichi Manyama, May 23 2023

Crossrefs

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

Cf. A340601, A340692.

Sequence in context: A162157 A060210 A260460 * 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