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 A261119 Expansion of f(x^2, -x^4) * f(x, x^5)^2 / f(-x^12, -x^12) in powers of x where f(, ) is Ramanujan's general theta function. 4
 1, 2, 2, 2, 0, 0, 1, 2, 4, 0, 0, 0, 0, 4, 2, 2, 0, 0, 3, 2, 2, 2, 0, 0, 2, 2, 2, 0, 0, 0, 0, 2, 2, 2, 0, 0, 3, 2, 4, 2, 0, 0, 0, 6, 2, 0, 0, 0, 0, 2, 4, 0, 0, 0, 2, 2, 2, 4, 0, 0, 1, 0, 2, 0, 0, 0, 0, 2, 6, 2, 0, 0, 2, 4, 0, 2, 0, 0, 4, 4, 0, 0, 0, 0, 0, 4, 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..10000 Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of f(x^2, x^6) * f(x, x^5)^2 / f(x^4, x^8) in powers of x where f(,) is Ramanujan'sgeneral theta function. Expansion of q^(-3/4) * eta(q^2)^3 * eta(q^3)^2 * eta(q^4) * eta(q^24) / (eta(q)^2 * eta(q^6)^2 * eta(q^8)) in powers of q. Euler transform of period 24 sequence [ 2, -1, 0, -2, 2, -1, 2, -1, 0, -1, 2, -2, 2, -1, 0, -1, 2, -1, 2, -2, 0, -1, 2, -2, ...]. a(n) = (-1)^n * A257921(n) = A129402(2*n + 1) = A261118(3*n + 2) = A192013(4*n + 3) = A000377(4*n + 3). a(2*n) = A257920(n). a(2*n + 1) = 2 * A259896(n). a(3*n) = A261118(n). EXAMPLE G.f. = 1 + 2*x + 2*x^2 + 2*x^3 + x^6 + 2*x^7 + 4*x^8 + 4*x^13 + 2*x^14 + ... G.f. = q^3 + 2*q^7 + 2*q^11 + 2*q^15 + q^27 + 2*q^31 + 4*q^35 + 4*q^55 + ... MATHEMATICA a[ n_] := If[ n < 0, 0, With[ {m = 4 n + 3}, (-1)^n DivisorSum[ m, KroneckerSymbol[ 12, #] KroneckerSymbol[ -2, m/#] &]]]; (* Michael Somos, Dec 22 2016 *) PROG (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^24 + A) / (eta(x + A)^2 * eta(x^6 + A)^2 * eta(x^8 + A)), n))}; (PARI) a(n) = my(m = 4*n+3); (-1)^n*sumdiv(m, d, kronecker(12, d) * kronecker(-2, m/d)); \\ Michel Marcus, Dec 13 2017 CROSSREFS Cf. A000377, A129402, A192013, A257920, A257921, A259896, A261118. Sequence in context: A132896 A089789 A257921 * A004541 A178045 A321300 Adjacent sequences: A261116 A261117 A261118 * A261120 A261121 A261122 KEYWORD nonn AUTHOR Michael Somos, Aug 08 2015 STATUS approved

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Last modified December 6 08:36 EST 2022. Contains 358605 sequences. (Running on oeis4.)