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A143062
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Expansion of false theta series variation of Euler's pentagonal number series in powers of x.
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8
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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
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OFFSET
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0,1
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COMMENTS
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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).
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REFERENCES
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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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, see p. 41, 10th equation numerator.
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LINKS
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FORMULA
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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)=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 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)
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EXAMPLE
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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 + ...
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MATHEMATICA
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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 *)
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PROG
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(PARI) {a(n) = if( issquare( 24*n + 1, &n), (-1)^(n \ 3) )};
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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