The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A129391 Expansion of phi(-x) * phi(x^5) / (chi(-x^2) * chi(-x^10)) in powers of x where phi(), chi() are Ramanujan theta functions. 4
 1, -2, 1, -2, 3, 0, 0, -2, 0, 0, 4, -2, 1, -4, 2, 0, 0, -2, 0, 0, 2, -2, 3, -2, 3, 0, 0, 0, 0, 0, 2, -6, 0, -2, 4, 0, 0, -2, 0, 0, 5, -2, 0, -4, 2, 0, 0, 0, 0, 0, 2, -2, 4, -2, 2, 0, 0, -2, 0, 0, 1, -4, 1, -2, 4, 0, 0, -4, 0, 0, 4, 0, 2, -6, 2, 0, 0, 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). 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 q^(-1/2) * eta(q)^2 * eta(q^4) * eta(q^10)^4 / (eta(q^2)^2 * eta(q^5)^2 * eta(q^20)) in powers of q. Euler transform of period 20 sequence [ -2, 0, -2, -1, 0, 0, -2, -1, -2, -2, -2, -1, -2, 0, 0, -1, -2, 0, -2, -2, ...]. a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0, b(5^e) = 1, b(p^e) = (-1)^e* (e+1) if p == 3, 7 (mod 20), b(p^e) = e+1 if p == 1, 9 (mod 20), b(p^e) = (1+(-1)^e)/2 if p == 11, 13, 17, 19 (mod 20). G.f.: Sum_{k>0} a(k) * x^(2*k - 1) = Sum_{k>0} (-1)^k * f(x^(2*k - 1)) where f(x) := x * (1 - x^2) * (1 - x^6) / (1 - x^10). a(n) = (-1)^n * A129390(n). a(n) = A111949(2*n + 1) = A143323(2*n). G.f. is a period 1 Fourier series which satisfies f(-1 / (40 t)) = 20^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A143323. EXAMPLE G.f. = 1 - 2*x + x^2 - 2*x^3 + 3*x^4 - 2*x^7 + 4*x^10 - 2*x^11 + x^12 - 4*x^13 + ... G.f. = q - 2*q^3 + q^5 - 2*q^7 + 3*q^9 - 2*q^15 + 4*q^21 - 2*q^23 + q^25 - ... MATHEMATICA a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 2 n + 1, KroneckerSymbol[ -20, #]&]]; (* Michael Somos, Nov 12 2015 *) PROG (PARI) {a(n) = if( n<0, 0, (-1)^n * sumdiv(2*n + 1, d, kronecker( -20, d)))}; (PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 2*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==5, 1, p%20 <10, (-1)^( ((p%20)%4 == 3)*e) * (e+1), 1-e%2 )))}; (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) * eta(x^10 + A)^4 / (eta(x^2 + A)^2 * eta(x^5 + A)^2 * eta(x^20 + A)), n))}; CROSSREFS Cf. A129390, A143323. Sequence in context: A127510 A328362 A158810 * A129390 A345255 A348910 Adjacent sequences: A129388 A129389 A129390 * A129392 A129393 A129394 KEYWORD sign AUTHOR Michael Somos, Apr 13 2007 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 24 00:02 EDT 2024. Contains 373661 sequences. (Running on oeis4.)