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 A122865 Expansion of chi(x) * phi(x^3) * psi(-x^3) in powers of x where chi(), phi(), psi() are Ramanujan theta functions. 31
 1, 1, 0, 2, 2, 1, 0, 0, 3, 0, 0, 2, 2, 2, 0, 0, 1, 2, 0, 2, 2, 1, 0, 0, 2, 0, 0, 2, 4, 0, 0, 0, 2, 3, 0, 2, 2, 0, 0, 0, 1, 0, 0, 4, 0, 2, 0, 0, 4, 2, 0, 0, 2, 2, 0, 0, 3, 0, 0, 2, 2, 0, 0, 0, 2, 1, 0, 2, 4, 2, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 2, 2, 0, 4, 0, 1, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS RRamanujan 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) * f(x^3) * f(-x^6) in powers of x where chi(), f() are Ramanujan theta functions. - Michael Somos, Sep 02 2015 Expansion of q^(-1/3) * eta(q^2)^2 * et(q^6)^4 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12)) in powers of q. Euler transform of period 12 sequence [1, -1, 2, 0, 1, -4, 1, 0, 2, -1, 1, -2, ...]. - Michael Somos, Apr 19 2007 G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A258228. - Michael Somos, Sep 02 2015 a(n) = A002654(3*n + 1) = A035154(3*n + 1) = A113446(3*n + 1) = A122864(3*n + 1) = A163746(3*n + 1). a(n) = (-1)^n * A258277(n). a(2*n + 1) = A122856(n). - Michael Somos, Sep 02 2015 a(4*n) = A002175(n). a(4*n + 2) = 0. - Michael Somos, Jan 19 2017 EXAMPLE G.f. = 1 + x + 2*x^3 + 2*x^4 + x^5 + 3*x^8 + 2*x^11 + 2*x^12 + 2*x^13 + ... G.f. = q + q^4 + 2*q^10 + 2*q^13 + q^16 + 3*q^25 + 2*q^34 + 2*q^37 + ... MATHEMATICA phi[q_] := EllipticTheta[3, 0, q]; chi[q_] := ((1 - InverseEllipticNomeQ[q]) * InverseEllipticNomeQ[q]/(16*q))^(-1/24); psi[q_] := (1/2)*q^(-1/8)*EllipticTheta[ 2, 0, q^(1/2)]; s = Series[ chi[q]*phi[q^3]*psi[-q^3], {q, 0, 104}]; a[n_] := Coefficient[s, q, n]; (* or *) a[n_] := If[n == 0, 1, Sum[Boole[Mod[d, 4] == 1] - Boole[Mod[d, 4] == 3], {d, Divisors[3n+1]}]]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Feb 17 2015, after PARI code *) a[ n_] := If[ n < 0, 0, DivisorSum[ 3 n + 1, KroneckerSymbol[ -4, #] &]]; (* Michael Somos, Sep 02 2015 *) a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^3] QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Sep 02 2015 *) PROG (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^6 + A)^4 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)), n))}; (PARI) {a(n) = my(A, p, e); if(n <0, 0, n = 3*n+1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 1, p==3, -2*(-1)^e, p%4==1, e+1, 1-e%2)))}; (PARI) {a(n) = if( n<0, 0, n = 3*n+1; sumdiv(n, d, (d%4==1) - (d%4==3)))}; /* Michael Somos, Apr 19 2007 */ CROSSREFS Cf. A002175, A002654, A035154, A113446, A122856, A122864, A163746, A258277. Sequence in context: A108040 A137566 A258277 * A246656 A352988 A334493 Adjacent sequences: A122862 A122863 A122864 * A122866 A122867 A122868 KEYWORD nonn AUTHOR Michael Somos, Sep 15 2006 STATUS approved

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Last modified January 28 20:13 EST 2023. Contains 359905 sequences. (Running on oeis4.)