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

%I A000025 M0433 N0164 #88 Feb 16 2025 08:32:18
%S A000025 1,1,-2,3,-3,3,-5,7,-6,6,-10,12,-11,13,-17,20,-21,21,-27,34,-33,36,
%T A000025 -46,51,-53,58,-68,78,-82,89,-104,118,-123,131,-154,171,-179,197,-221,
%U A000025 245,-262,279,-314,349,-369,398,-446,486,-515,557,-614,671,-715,767,-845,920,-977,1046,-1148,1244
%N A000025 Coefficients of the 3rd-order mock theta function f(q).
%C A000025 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 A000025 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 82, Examples 4 and 5.
%D A000025 Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355
%D A000025 Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 17, 31.
%D A000025 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000025 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000025 Vaclav Kotesovec, <a href="/A000025/b000025.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe and then corrected by Sean A. Irvine, Apr 25 2019)
%H A000025 George E. Andrews, <a href="http://www.jstor.org/stable/2321943">An introduction to Ramanujan's "lost" notebook</a>, Amer. Math. Monthly 86 (1979), no. 2, 89-108.  See page 95.
%H A000025 Steven Charlton, <a href="https://arxiv.org/abs/2412.17459">Explicit linear dependence congruence relations for the partition function modulo 4</a>, arXiv:2412.17459 [math.NT], 2024. See p. 3.
%H A000025 Leila 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 A000025 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 [math.RT], 2015. [See f(q)]
%H A000025 Nathan 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 A000025 Ken 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 A000025 George N. Watson, <a href="https://doi.org/10.1112/jlms/s1-11.1.55">The final problem: an account of the mock theta functions</a>, J. London Math. Soc., 11 (1936) 55-80.
%H A000025 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MockThetaFunction.html">Mock Theta Function</a>.
%F A000025 G.f.: 1 + Sum_{n>=1} (q^(n^2) / Product_{i=1..n} (1 + q^i)^2).
%F A000025 G.f.: (1 + 4 * Sum_{n>=1} (-1)^n * q^(n*(3*n+1)/2) / (1 + q^n)) / Product_{i>=1} (1 - q^i).
%F A000025 a(n) ~ -(-1)^n * exp(Pi*sqrt(n/6)) / (2*sqrt(n)) [Ramanujan]. - _Vaclav Kotesovec_, Jun 10 2019
%F A000025 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
%F A000025 From _Seiichi Manyama_, May 23 2023: (Start)
%F A000025 a(n) = A340601(n) - A340692(n).
%F A000025 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)
%e A000025 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 A000025 a:= m-> coeff(series((1+4*add((-1)^n*q^(n*(3*n+1)/2)/
%p A000025         (1+q^n), n=1..m))/mul(1-q^i, i=1..m), q, m+1), q, m):
%p A000025 seq(a(n), n=0..120);
%t A000025 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 A000025 sgn[P_ (* a partition *)] :=
%t A000025 Signature[
%t A000025   PermutationList[
%t A000025    Cycles[Flatten[
%t A000025      SplitBy[Range[Total[P]], (Function[{x}, x > #1] &) /@
%t A000025        Accumulate[P]], Length[P] - 1]]]]
%t A000025 conjugate[P_List(* a partition *)] :=
%t A000025 Module[{s = Select[P, #1 > 0 &], i, row, r}, row = Length[s];
%t A000025   Table[r = row; While[s[[row]] <= i, row--]; r, {i, First[s]}]]
%t A000025 Total[Function[{x}, sgn[x] sgn[conjugate[x]]] /@
%t A000025     IntegerPartitions[#]] & /@ Range[20]
%t A000025 (* _George Beck_, Oct 25 2014 *)
%t A000025 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 *)
%t A000025 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 *)
%o A000025 (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 */
%o A000025 (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
%Y A000025 Other '3rd-order' mock theta functions are at A013953, A053250, A053251, A053252, A053253, A053254, A053255. See also A000039, A000199.
%Y A000025 Cf. A340601, A340692.
%K A000025 sign,easy,nice
%O A000025 0,3
%A A000025 _N. J. A. Sloane_
%E A000025 Entry improved by comments from _Dean Hickerson_