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

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

Author

N. J. A. Sloane

Keywords

• sign
• easy
• nice

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.

Examples

			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 + ...
		

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• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 17, 31.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)
• George E. Andrews, An introduction to Ramanujan's "lost" notebook, Amer. Math. Monthly 86 (1979), no. 2, 89-108. See page 95.
• Steven Charlton, Explicit linear dependence congruence relations for the partition function modulo 4, arXiv:2412.17459 [math.NT], 2024. See p. 3.
• Leila 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)]
• Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 55, Eq. (26.11), (26.24).
• Ken 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.

Crossrefs

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

Cf. A340601, A340692.

Programs

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 *)
rnk[prts_]:=Max[prts]-Length[prts]; mtf[n_]:=Module[{pn=IntegerPartitions[n]},Total[If[ EvenQ[ rnk[#]],1,-1]&/@pn]]; Join[{1},Array[mtf,60]] (* Harvey P. Dale, Sep 13 2024 *)

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

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)

Extensions

Entry improved by comments from Dean Hickerson