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 A199918 Expansion of false theta series variation of Euler's pentagonal number series in powers of x. 3
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 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. 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)|. EXAMPLE 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, KroneckerSymbol[ -6, Sqrt[ 24 n + 1]], 0]; PROG (PARI) {a(n) = my(m); if( issquare( 24*n + 1, &m), kronecker( -6, m), 0)}; CROSSREFS Cf. A010815, A143062. Sequence in context: A206959 A080995 A121373 * A229894 A256538 A074910 Adjacent sequences: A199915 A199916 A199917 * A199919 A199920 A199921 KEYWORD sign AUTHOR Michael Somos, Nov 12 2011 STATUS approved

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Last modified December 8 08:02 EST 2023. Contains 367662 sequences. (Running on oeis4.)