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A199918 Expansion of false theta series variation of Euler's pentagonal number series in powers of x. 3

%I #24 Sep 07 2018 01:46:31

%S 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 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,

%U 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

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

%H G. C. Greubel, <a href="/A199918/b199918.txt">Table of n, a(n) for n = 0..10000</a>

%H 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.

%F 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.

%F 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).

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

%e G.f. = 1 + x + x^2 + x^5 - x^7 - x^12 - x^15 - x^22 + x^26 + x^35 + x^40 + ...

%e G.f. = q + q^25 + q^49 + q^121 - q^169 - q^289 - q^361 - q^529 + q^625 + q^841 + ...

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

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

%Y Cf. A010815, A143062.

%K sign

%O 0,1

%A _Michael Somos_, Nov 12 2011

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