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A143062 Expansion of false theta series variation of Euler's pentagonal number series in powers of x.

%I A143062 #50 Feb 01 2025 00:49:40
%S A143062 1,-1,1,0,0,-1,0,1,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,
%T A143062 0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,
%U A143062 0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0
%N A143062 Expansion of false theta series variation of Euler's pentagonal number series in powers of x.
%C A143062 a(n) = sum over all partitions of n into distinct parts of number of partitions with even largest part minus number with odd largest part.
%C A143062 In the Berndt reference replace {a -> 1, q -> x} in equation (3.1) to get g.f. Replace {a -> x, q -> x} to get f(x). G.f. is 1 - f(x) * x / (1 + x).
%D A143062 G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See Section 9.4, pp. 232-236.
%D A143062 Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, see p. 41, 10th equation numerator.
%H A143062 G. C. Greubel, <a href="/A143062/b143062.txt">Table of n, a(n) for n = 0..1000</a>
%H A143062 B. C. Berndt, B. Kim, and A. J. Yee, <a href="http://dx.doi.org/10.1016/j.jcta.2009.07.005">Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions</a>, J. Comb. Thy. Ser. A, 117 (2010), 957-973.
%H A143062 I. Pak, <a href="http://www.math.ucla.edu/~pak/papers/finefull.pdf">On Fine's partition theorems, Dyson, Andrews and missed opportunities</a>, Math. Intelligencer, 25 (No. 1, 2003), 10-16.
%F A143062 a(n) = b(24*n + 1) where b() is multiplicative with b(p^(2*e)) = (-1)^e if p = 5 (mod 6), b(p^(2*e)) = +1 if p = 1 (mod 6) and b(p^(2*e-1)) = b(2^e) = b(3^e) = 0 if e>0.
%F A143062 G.f.: Sum_{k>=0} x^((3*k^2 + k) / 2) * (1 - x^(2*k + 1)) = 1 - Sum_{k>0} x^((3*k^2 - k) / 2) * (1 - x^k).
%F A143062 G.f.: 1 - x / (1 + x) + x^3 / ((1 + x) * (1 + x^2)) - x^6 / ((1 + x) * (1 + x^2) * (1 + x^3)) + ...
%F A143062 G.f.: 1 - x / (1 + x^2) + x^2 / ((1 + x^2) * (1 + x^4)) - x^3 / ((1 + x^2 ) * (1 + x^4) * (1 + x^6)) + ...
%F A143062 |a(n)| = |A010815(n)| = |A080995(n)| = |A199918(n)| = |A121373(n)|.
%F A143062 From _Joerg Arndt_, Jun 24 2013: (Start)
%F A143062 a(n) = A026838(n) - A026837(n) (Fine's theorem), see the Pak reference.
%F A143062 a(n)=1 if n = k(3k+1)/2, a(n)=-1 if n = k(3k-1)/2, a(n)=0 otherwise.
%F A143062 G.f.: Sum_{n >= 0} (-q)^n * (Product_{k = 1..n-1} 1 + q^k). (End)
%F A143062 a(n) = - A203568(n) unless n=0. a(0) = 1. - _Michael Somos_, Jul 12 2015
%F A143062 From _Peter Bala_, Feb 04 2021: (Start)
%F A143062 A conjectural g.f: 1 + Sum_{n >= 0} (-1)^n*x^(2*n-1)/Product_{k = 1..n} 1 + x^(2*k-1).
%F A143062 G.f.: 1 - Sum_{n >= 1} x^(n*(2*n-1))/Product_{k = 1..2*n} 1 + x^k [added Dec 19 2024: see Berndt et al., Entry 9.44]. (End)
%F A143062 Conjectural g.f.: (1/(1 + x)) * (2 - Sum_{n >= 0} (-1)^n * x^(3*n)/Product_{k = 1..n} 1 + x^(2*k)). - _Peter Bala_, Jan 19 2025
%e A143062 a(5) = -1 +1 -1 = -1 since 5 = 4 + 1 = 3 + 2. a(7) = -1 +1 -1 +1 +1 = 1 since 7 = 6 + 1 = 5 + 2 = 4 + 3 = 4 + 2 + 1.
%e A143062 G.f. = 1 - x + x^2 - x^5 + x^7 - x^12 + x^15 - x^22 + x^26 - x^35 + x^40 + ...
%e A143062 G.f. = q - q^25 + q^49 - q^121 + q^169 - q^289 + q^361 - q^529 + q^625 - q^841 + ...
%t A143062 a[ n_] := If[ SquaresR[ 1, 24 n + 1] == 2, (-1)^Quotient[ Sqrt[24 n + 1], 3], 0];
%t A143062 a[ n_] := With[ {m = Sqrt[24 n + 1]}, If[ IntegerQ@m, (-1)^Quotient[ m, 3], 0]]; (* _Michael Somos_, Nov 18 2015 *)
%o A143062 (PARI) {a(n) = if( issquare( 24*n + 1, &n), (-1)^(n \ 3) )};
%Y A143062 Cf. A010815, A026837, A026838, A080995, A121373, A199918, A203568, A268539.
%K A143062 sign,easy
%O A143062 0,1
%A A143062 _Michael Somos_, Jul 21 2008