A257656 Expansion of f(x) * f(x^3) * f(-x^4)^2 * chi(-x^6)^2 in powers of x where chi(), f() are Ramanujan theta functions.
1, 1, -1, 1, -1, -4, -1, -6, -1, 1, 4, 12, -1, 14, 6, -4, -1, -16, -1, -18, 4, -6, -12, 24, -1, 21, -14, 1, 6, -28, 4, -30, -1, 12, 16, 24, -1, 38, 18, 14, 4, -40, 6, -42, -12, -4, -24, 48, -1, 43, -21, -16, -14, -52, -1, -48, 6, -18, 28, 60, 4, 62, 30, -6, -1
Offset: 0
Examples
G.f. = 1 + x - x^2 + x^3 - x^4 - 4*x^5 - x^6 - 6*x^7 - x^8 + x^9 + 4*x^10 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..10000
- Michael Somos, Introduction to Ramanujan theta functions, 2019.
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Programs
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Mathematica
a[ n_] := SeriesCoefficient[ QPochhammer[ -x] QPochhammer[ -x^3] QPochhammer[ x^4]^2 QPochhammer[ x^6, x^12]^2, {x, 0, n}]; a[ n_] := If[ n<1, Boole[n == 0], Times @@ (If[ # < 5, -(-1)^#, With[ {t = # KroneckerSymbol[ 12, #]}, (t^(#2 + 1) - 1 ) / (t - 1)]]& @@@ FactorInteger @ n)]; -
PARI
{a(n) = if( n<1, n==0, -sumdiv(n, d, d * kronecker( 12, d) * (-1)^(n/d)))}; -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A) * eta(x^6 + A)^5 / (eta(x + A) * eta(x^3 + A) * eta(x^12 + A)^3), n))}; -
PARI
{a(n) = my(A, p, e, t); if( n<1, n==0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, -(-1)^p, t = p * kronecker( 12, p); (t^(e+1) - 1) / (t - 1))))};
Formula
Expansion of eta(q^2)^3 * eta(q^4) * eta(q^6)^5 / (eta(q) * eta(q^3) * eta(q^12)^3) in powers of q.
Euler transform of period 12 sequence [1, -2, 2, -3, 1, -6, 1, -3, 2, -2, 1, -4, ...].
Multiplicative with a(2^e) = -1, a(p^e) = ((p*Kronecker(12, p))^(e+1) - 1)/(p*Kronecker(12, p) - 1) for odd prime p. - Andrew Howroyd, Jul 27 2018
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(24*sqrt(3)) = 0.237425... . - Amiram Eldar, Jan 29 2024
Comments