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 A259761 Expansion of (phi(x)^2 + phi(x^9)^2) / 2 in powers of x where phi() is a Ramanujan theta function. 4
 1, 2, 2, 0, 2, 4, 0, 0, 2, 4, 4, 0, 0, 4, 0, 0, 2, 4, 4, 0, 4, 0, 0, 0, 0, 6, 4, 0, 0, 4, 0, 0, 2, 0, 4, 0, 4, 4, 0, 0, 4, 4, 0, 0, 0, 8, 0, 0, 0, 2, 6, 0, 4, 4, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 2, 8, 0, 0, 4, 0, 0, 0, 4, 4, 4, 0, 0, 0, 0, 0, 4, 4, 4, 0, 0, 8, 0 (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 phi(x) = 1 + 2*Sum_{m=1..oo} x^(m^2). - N. J. A. Sloane, Jan 30 2017 Expansion of phi(x) * phi(x^9) + 2 * x^2 * chi(x^3)^2 * psi(-x^9)^2 in powers of x where phi(), psi(), chi() are Ramanujan theta functions. a(n) = 2 * b(n) with a(0) = 1 and b() is multiplicative with b(2^e) = 1, b(3^e) = 1 + (-1)^e if e>0, b(p^e) = e+1 if p == 1, 5 (mod 12), (p^e) = (1 + (-1)^e)/2 if p == 7, 11 (mod 12). a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0. a(2*n) = a(n). a(9*n) = A004018(n). a(6*n + 4) = 2 * A122856(n). EXAMPLE G.f. = 1 + 2*x + 2*x^2 + 2*x^4 + 4*x^5 + 2*x^8 + 4*x^9 + 4*x^10 + 4*x^13 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x]^2 + EllipticTheta[ 3, 0, x^9]^2) / 2, {x, 0, n}]; a[ n_] := If[ n < 1, Boole[n == 0], 2 Times @@ ( Which[ # < 3, 1, # == 3, 1 + (-1)^#2, Mod[#, 12] < 6, #2 + 1, True, (1 + (-1)^#2) / 2 ] & @@@ FactorInteger[n])]; PROG (PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 1, p==3, 1 + (-1)^e, p%12>6, (1 + (-1)^e) / 2, e+1)))}; (MAGMA) A := Basis( ModularForms( Gamma1(36), 1), 87); A[1] + 2*A[2] + 2*A[3] + 2*A[5] + 4*A[6] + 2*A[9] + 4*A[10] + 4*A[11] + 4*A[14] + 2*A[17] + 4*A[18] + 4*A[19]; CROSSREFS Cf. A004018, A122856. Sequence in context: A141058 A102706 A105673 * A171933 A074823 A226177 Adjacent sequences:  A259758 A259759 A259760 * A259762 A259763 A259764 KEYWORD nonn AUTHOR Michael Somos, Jul 04 2015 STATUS approved

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Last modified May 15 06:46 EDT 2021. Contains 343909 sequences. (Running on oeis4.)