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 A028625 Expansion of (theta_3(z)*theta_3(15z)+theta_2(z)*theta_2(15z)). 2
 1, 2, 0, 0, 6, 0, 4, 0, 0, 2, 4, 0, 0, 0, 0, 2, 10, 0, 0, 4, 0, 0, 0, 0, 8, 2, 0, 0, 0, 0, 0, 4, 0, 0, 8, 0, 6, 0, 0, 0, 8, 0, 0, 0, 0, 0, 8, 0, 0, 2, 0, 4, 0, 0, 4, 0, 0, 0, 0, 0, 6, 4, 0, 0, 14, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 12, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 8, 0, 12, 0, 0, 0, 6, 0, 0, 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). Theta series of quadratic form (or lattice) with Gram matrix [ 2, 1; 1, 8 ]. The number of integer solutions (x, y) to x^2 + x*y + 4*y^2 = n, discriminant -15. - Ray Chandler, Jul 12 2014 a(n) = number of solutions in integers (x, y) of x^2 + 15*y^2 = 4*n. - Michael Somos, Jul 17 2018 LINKS John Cannon, Table of n, a(n) for n = 0..5000 N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references) Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of phi(q) * phi(q^15) + 4 * q^4 * psi(q^2) * psi(q^30) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Aug 26 2006 Expansion of (eta(q^3) * eta(q^5))^2 / (eta(q)* eta(q^15)) + (eta(q)  *eta(q^15))^2 / (eta(q^3) * eta(q^5)) in powers of q. - Michael Somos, Aug 26 2006 a(n) = A260671(4*n). - Michael Somos, Jul 17 2018 EXAMPLE G.f. = 1 + 2*q^2 + 6*q^8 + 4*q^12 + 2*q^18 + 4*q^20 + 2*q^30 + 10*q^32 + 4*q^38 + 8*q^48 + 2*q^50 + 4*q^62 + 8*q^68 + 6*q^72 + 8*q^80 + 8*q^92 + 2*q^98 + ... G.f. = 1 + 2*x + 6*x^4 + 4*x^6 + 2*x^9 + 4*x^10 + 2*x^15 + 10*x^16 + 4*x^19 + ... - Michael Somos, Jul 17 2018 MATHEMATICA r[n_] := Reduce[x^2 + x*y + 4*y^2 == n, {x, y}, Integers]; Table[rn = r[n]; Which[rn === False, 0, Head[rn] === Or, Length[rn], Head[rn] === And, 1], {n, 0, 105}] (* Jean-François Alcover, Nov 05 2015, after the comment by Ray Chandler *) a[0] = 1; a[n_] := With[{K = KroneckerSymbol}, DivisorSum[n, K[-15, #] + K[#, 3]*K[n/#, 5]&]]; Table[a[n], {n, 0, 103}] (* Jean-François Alcover, Jul 07 2017, after Michael Somos *) a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 3, 0, x^15] + EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, x^15], {x, 0, n}]; (* Michael Somos, Jul 17 2018 *) PROG (PARI) {a(n) = if( n<1, n==0, qfrep([2, 1; 1, 8], n, 1)[n]*2)}; /* Michael Somos, Aug 26 2006 */ (PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, kronecker(-15, d) + kronecker(d, 3) * kronecker(n/d, 5) ))}; /* Michael Somos, Aug 26 2006 */ CROSSREFS Cf. A260671. Sequence in context: A241020 A277444 A274710 * A344441 A221728 A339942 Adjacent sequences:  A028622 A028623 A028624 * A028626 A028627 A028628 KEYWORD nonn AUTHOR STATUS approved

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Last modified November 26 21:07 EST 2021. Contains 349344 sequences. (Running on oeis4.)