# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a253625 Showing 1-1 of 1 %I A253625 #21 Jun 12 2024 03:29:40 %S A253625 1,-1,3,-1,3,0,3,-2,3,-1,0,0,3,-2,6,0,3,0,3,-2,0,-2,0,0,3,-1,6,-1,6,0, %T A253625 0,-2,3,0,0,0,3,-2,6,-2,0,0,6,-2,0,0,0,0,3,-3,3,0,6,0,3,0,6,-2,0,0,0, %U A253625 -2,6,-2,3,0,0,-2,0,0,0,0,3,-2,6,-1,6,0,6,-2 %N A253625 Expansion of psi(q^2) * f(-q, q^2)^2 / f(-q, -q^5) in powers of q where psi(), f() are Ramanujan theta functions. %C A253625 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). %C A253625 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882). %H A253625 G. C. Greubel, Table of n, a(n) for n = 0..1000 %H A253625 Michael Somos, Introduction to Ramanujan theta functions %H A253625 Eric Weisstein's World of Mathematics, Ramanujan Theta Functions. %F A253625 Expansion of psi(q^2)^2 * phi(q^3)^2 / (psi(q) * psi(q^3)) = f(-q) * f(-q^3) * (chi(q^3) / chi(-q^2))^4 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. %F A253625 Expansion of (-a(q) - 3*a(q^2) + 4*a(q^4)) / 6 = b(q^4) * (b(q) + 2*b(q^4)) / (3*b(q^2)) in powers of q where a(), b() are cubic AGM theta functions. %F A253625 Expansion of eta(q) * eta(q^4)^4 * eta(q^6)^8 / (eta(q^2)^4 * eta(q^3)^3 * eta(q^12)^4) in powers of q. %F A253625 Euler transform of period 12 sequence [ -1, 3, 2, -1, -1, -2, -1, -1, 2, 3, -1, -2, ...]. %F A253625 Moebius transform is period 12 sequence [ -1, 4, 0, 0, 1, 0, -1, 0, 0, -4, 1, 0, ...]. %F A253625 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 48^(-1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A253623. %F A253625 a(n) = -b(n) where b() is multiplicative with b(2^e) = -3 if e>0, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6). %F A253625 G.f.: 1 + Sum_{k>0} (3 - (k mod 2)*4) * (x^k + x^(3*k)) / (1 + x^(2*k) + x^(4*k)). %F A253625 G.f.: Product_{k>0} (1 - x^k) * (1 - x^(3*k)) * (1 + x^(3*k))^4 / (1 - x^(2*k) + x^(4*k))^4. %F A253625 a(n) = (-1)^n * A253626(n). a(2*n) = A107760(n). a(2*n + 1) = - A033762(n). a(3*n) = a(n). a(3*n + 1) = - A122861(n). a(4*n + 1) = - A112604(n). a(4*n + 2) = 3 * A033762(n). a(4*n + 3) = - A112605(n). %F A253625 a(6*n + 1) = - A097195(n). a(6*n + 2) = 3 * A033687(n). a(6*n + 5) = 0. a(12*n + 1) = - A123884(n). a(12*n + 7) = -2 * A121361(n). a(12*n + 10) = 0. %e A253625 G.f. = 1 - q + 3*q^2 - q^3 + 3*q^4 + 3*q^6 - 2*q^7 + 3*q^8 - q^9 + 3*q^12 + ... %t A253625 a[ n_] := If[ n < 1, Boole[n == 0], (-1)^ n Sum[(-1)^ Quotient[ d, 3] {1, 1, 0}[[ Mod[d, 3, 1] ]] {1, 2}[[ Mod[n/d, 2, 1] ]], {d, Divisors @ n}]]; %t A253625 a[ n_] := SeriesCoefficient[ QPochhammer[ q] QPochhammer[ q^3] (QPochhammer[ -q^3, q^6] QPochhammer[ -q^2, q^2])^4, {q, 0, n}]; %t A253625 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3]^2 EllipticTheta[ 2, 0, q]^2 / (EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 2, 0, q^(3/2)]), {q, 0, n}]; %o A253625 (PARI) {a(n) = if( n<1, n==0, (-1)^n * sumdiv(n, d, (-1)^(d\3) * (d%3>0) * (2-(n\d)%2)))}; %o A253625 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^4 * eta(x^6 + A)^8 / (eta(x^2 + A)^4 * eta(x^3 + A)^3 * eta(x^12 + A)^4), n))}; %o A253625 (PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); - prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, -3, if( p==3, 1, if( p%6 == 1, e+1, 1-e%2))))))}; %o A253625 (Magma) A := Basis( ModularForms( Gamma1(12), 1), 81); A[1] - A[2] + 3*A[3] - A[4] + 3*A[5]; %Y A253625 Cf. A033687, A033762, A097195, A107760, A112604, A112605, A121361, A122861, A123884, A253623, A253626. %K A253625 sign %O A253625 0,3 %A A253625 _Michael Somos_, Jan 06 2015 %E A253625 Typo in formula fixed by _Colin Barker_, Jan 08 2015 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE