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 A122856 Expansion of f(x, x^5)^2 in powers of x where f(, ) is Ramanujan's general theta function. 33
 1, 2, 1, 0, 0, 2, 2, 0, 2, 2, 1, 0, 0, 2, 0, 0, 3, 2, 0, 0, 0, 4, 2, 0, 2, 0, 2, 0, 0, 2, 0, 0, 1, 2, 2, 0, 0, 2, 2, 0, 2, 4, 1, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0, 4, 2, 0, 0, 0, 4, 0, 0, 2, 2, 3, 0, 0, 0, 2, 0, 2, 4, 0, 0, 0, 2, 0, 0, 1, 2, 0, 0, 0, 2, 4, 0, 0, 2, 2, 0, 0, 2, 0, 0, 4, 2, 2, 0, 0, 4, 0, 0, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of (chi(x) * psi(-x^3))^2 in powers of x where chi(), psi() are Ramanujan theta functions. Expansion of q^(-2/3) * (eta(q^2)^2 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6)))^2 in powers of q. Euler transform of period 12 sequence [2, -2, 0, 0, 2, -2, 2, 0, 0, -2, 2, -2, ...]. a(4*n + 3) = a(8*n + 4) = 0. a(n) = A002654(3*n + 2) = A035154(3*n + 2) = A113446(3*n + 2). a(2*n) = A122865(n). a(4*n + 1) = 2 * A121444(n). a(4*n + 2) = A122856(n). a(n) = (-1)^n * A258278(n). Convolution square of A089801. EXAMPLE G.f. = 1 + 2*x + x^2 + 2*x^5 + 2*x^6 + 2*x^8 + 2*x^9 + x^10 + 2*x^13 + ... G.f. = q^2 + 2*q^5 + q^8 + 2*q^17 + 2*q^20 + 2*q^26 + 2*q^29 + q^32 + ... MATHEMATICA a[ n_] := If[ n < 0, 0, With[ {m = 3 n + 2}, Sum[ KroneckerSymbol[ -4, d], {d, Divisors@m}]]]; (* Michael Somos, Nov 14 2011 *) QP = QPochhammer; s = (QP[q^2]^2*QP[q^3]*(QP[q^12]/(QP[q]*QP[q^4]*QP[q^6]) ))^2 + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *) a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x^(1/3)] - EllipticTheta[ 3, 0, x^3])^2 / (4 x^(2/3)), {x, 0, n}]; (* Michael Somos, Jan 19 2017 *) a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x^2] EllipticTheta[ 2, Pi/4, x^(3/2)])^2 / (2 x^(3/4)), {x, 0, n}]; (* Michael Somos, Jan 19 2017 *) PROG (PARI) {a(n) = if( n<0, 0, n = 3*n+2; sumdiv(n, d, (d%4==1) - (d%4==3)))}; (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)))^2, n))}; CROSSREFS Cf. A002654, A035154, A089801, A113446, A121444, A122865, A258278. Sequence in context: A008626 A058626 A258278 * A328797 A328795 A353846 Adjacent sequences: A122853 A122854 A122855 * A122857 A122858 A122859 KEYWORD nonn AUTHOR Michael Somos, Sep 14 2006 STATUS approved

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Last modified February 8 21:33 EST 2023. Contains 360153 sequences. (Running on oeis4.)