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A143062
Expansion of false theta series variation of Euler's pentagonal number series in powers of x.
8
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, 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, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0
OFFSET
0,1
COMMENTS
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.
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).
REFERENCES
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.
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, see p. 41, 10th equation numerator.
LINKS
I. Pak, On Fine's partition theorems, Dyson, Andrews and missed opportunities, Math. Intelligencer, 25 (No. 1, 2003), 10-16.
FORMULA
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.
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).
G.f.: 1 - x / (1 + x) + x^3 / ((1 + x) * (1 + x^2)) - x^6 / ((1 + x) * (1 + x^2) * (1 + x^3)) + ...
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)) + ...
|a(n)| = |A010815(n)| = |A080995(n)| = |A199918(n)| = |A121373(n)|.
From Joerg Arndt, Jun 24 2013: (Start)
a(n) = A026838(n) - A026837(n) (Fine's theorem), see the Pak reference.
a(n)=1 if n = k(3k+1)/2, a(n)=-1 if n = k(3k-1)/2, a(n)=0 otherwise.
G.f.: Sum_{n >= 0} (-q)^n * (Product_{k = 1..n-1} 1 + q^k). (End)
a(n) = - A203568(n) unless n=0. a(0) = 1. - Michael Somos, Jul 12 2015
From Peter Bala, Feb 04 2021: (Start)
A conjectural g.f: 1 + Sum_{n >= 0} (-1)^n*x^(2*n-1)/Product_{k = 1..n} 1 + x^(2*k-1).
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)
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
EXAMPLE
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.
G.f. = 1 - x + x^2 - x^5 + x^7 - x^12 + x^15 - x^22 + x^26 - x^35 + x^40 + ...
G.f. = q - q^25 + q^49 - q^121 + q^169 - q^289 + q^361 - q^529 + q^625 - q^841 + ...
MATHEMATICA
a[ n_] := If[ SquaresR[ 1, 24 n + 1] == 2, (-1)^Quotient[ Sqrt[24 n + 1], 3], 0];
a[ n_] := With[ {m = Sqrt[24 n + 1]}, If[ IntegerQ@m, (-1)^Quotient[ m, 3], 0]]; (* Michael Somos, Nov 18 2015 *)
PROG
(PARI) {a(n) = if( issquare( 24*n + 1, &n), (-1)^(n \ 3) )};
KEYWORD
sign,easy,changed
AUTHOR
Michael Somos, Jul 21 2008
STATUS
approved