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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 (list; graph; refs; listen; history; text; internal format)
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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, see p. 41, 10th equation numerator.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)|.
a(n) = A026838(n) - A026837(n) (Fine's theorem), see the Pak reference. [Joerg Arndt, Jun 24 2013]
a(n)=1 if n = k(3k+1)/2, a(n)=-1 if n = k(3k-1)/2, a(n)=0 otherwise. [Joerg Arndt, Jun 24 2013]
G.f.: sum(n>=0, (-q)^n * prod(k=1..n-1, 1+q^k) ). [Joerg Arndt, Jun 24 2013]
a(n) = - A203568(n) unless n=0. a(0) = 1. - Michael Somos, Jul 12 2015
From Peter Bala, Feb 04 202: (Start)
A pair of conjectural g.f.s: 1 + Sum_{n >= 0} (-1)^n*x^(2*n-1)/Product_{k = 1..n} 1 + x^(2*k-1);
1 - Sum_{n >= 1} x^(n*(2*n-1))/Product_{k = 1..2*n} 1 + x^k. (End)
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) )};
CROSSREFS
Sequence in context: A133080 A316917 A133985 * A010815 A206958 A206959
KEYWORD
sign
AUTHOR
Michael Somos, Jul 21 2008
STATUS
approved

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Last modified March 29 08:13 EDT 2024. Contains 371265 sequences. (Running on oeis4.)