A199918 Expansion of false theta series variation of Euler's pentagonal number series in powers of x.

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, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0

Offset: 0

Author

Michael Somos, Nov 12 2011

Keywords

• sign

Examples

			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 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• B. C. Berndt, B. Kim, and A. J. Yee, Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions, J. Comb. Thy. Ser. A, 117 (2010), 957-973.

Crossrefs

Cf. A010815, A143062.

Programs

Mathematica

a[ n_] := If[ SquaresR[ 1, 24 n + 1] == 2, KroneckerSymbol[ -6, Sqrt[ 24 n + 1]], 0];

PARI

{a(n) = my(m); if( issquare( 24*n + 1, &m), kronecker( -6, m), 0)};

Formula

a(n) = b(24*n + 1) where b(n) is multiplicative with b(p^(2*e)) = (-1)^e if p == 13, 17, 29, 23 (mod 24), b(p^(2*e)) = +1 if p = 1, 5, 7, 11 (mod 24) and b(p^(2*e - 1)) = b(2^e) = b(3^e) = 0 if e > 0.

G.f.: 1 + Sum_{k>0} x^k / Product_{i=1..k} (1 + x^(2*i)) = 1 + Sum_{k>0} x^k * Product_{i=1..k-1} (1 + (-x)^i) = Sum_{k in Z} x^((k^2 - 1) / 24) * Kronecker(-24, k).

|a(n)| = |A010815(n)| = |A143062(n)|.

G.f.: A(x) = 1/(1 - x) * (2 - Sum_{k >= 0} x^(3*k)/Product_{i = 1..k} 1 + x^(2*i)) = 1/((1 - x)*(1 - x^3)) * (-2*x^3 + Sum_{k >= 0} x^(5*k)/Product_{i = 1..k} 1 + x^(2*i)). - Peter Bala, Jan 24 2025