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 A128582 Expansion of f(x^4, x^12) * f(x, x^5) in powers of x where f(, ) is Ramanujan's general theta function. 11
 1, 1, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 3, 1, 0, 0, 1, 3, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 2, 2, 0, 0, 1, 2, 0, 0, 2, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 2, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 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 psi(x) * psi(-x^3) / chi(-x^4)^2 in powers of x where psi(), chi() are Ramanujan theta functions. Expansion of q^(-5/6) * eta(q^2)^2 * eta(q^3) * eta(q^8)^2 * eta(q^12) / (eta(q) * eta(q^4)^2 * eta(q^6)) in powers of q. Euler transform of period 24 sequence [ 1, -1, 0, 1, 1, -1, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, -1, 1, 1, 0, -1, 1, -2, ...]. G.f.: Product_{k>0} (1 + x^k) * (1 - x^(2*k)) * (1 + x^(4*k))^2 * (1 - x^(6*k - 3)) * (1 - x^(12*k)). A128580(3*n + 2) = -2 * a(n). a(4*n) = A128583. a(4*n + 1) = A128591(n). a(4*n + 2) = a(4*n + 3) = 0. EXAMPLE G.f. = 1 + x + x^4 + 2*x^5 + x^8 + x^9 + 2*x^12 + x^13 + x^16 + x^17 + 2*x^20 + ... G.f. = q^5 + q^11 + q^29 + 2*q^35 + q^53 + q^59 + 2*q^77 + q^83 + q^101 + q^107 + ... MATHEMATICA a[ n_] := If[ n < 0, 0, With[{m = 6 n + 5}, -1/2 DivisorSum[ m, KroneckerSymbol[ -12, #] KroneckerSymbol[ 2, m/#] &]]]; (* Michael Somos, Nov 15 2015 *) a[ n_] := SeriesCoefficient[ 2^(-3/2) x^(-7/8) EllipticTheta[ 2, Pi/4, x^(3/2)] EllipticTheta[ 2, Pi/4, x]^2 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Nov 15 2015 *) PROG (PARI) {a(n) = if( n<0, 0, n = 6*n + 5; -1/2 * sumdiv( n, d, kronecker( -12, d) * kronecker( 2, n/d)))}; (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^8 + A)^2 * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A)), n))}; CROSSREFS Cf. A128580, A128583, A128591. Sequence in context: A224772 A025442 A260118 * A213185 A285716 A101606 Adjacent sequences:  A128579 A128580 A128581 * A128583 A128584 A128585 KEYWORD nonn AUTHOR Michael Somos, Mar 11 2007 STATUS approved

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Last modified July 29 05:47 EDT 2021. Contains 346340 sequences. (Running on oeis4.)