# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a263571 Showing 1-1 of 1 %I A263571 #15 Dec 28 2023 01:52:34 %S A263571 1,1,2,2,0,1,0,2,3,2,2,0,0,2,0,0,3,0,4,2,0,1,0,4,2,0,2,0,0,2,0,0,2,3, %T A263571 2,2,0,2,0,2,3,2,2,0,0,0,0,0,4,0,2,4,0,2,0,2,1,0,6,0,0,0,0,0,2,3,2,2, %U A263571 0,0,0,2,4,4,2,0,0,2,0,0,2,0,0,4,0,1,0 %N A263571 Expansion of f(x^2, x^2) * f(x, x^5) in powers of x where f(, ) is Ramanujan's general theta function. %C A263571 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). %H A263571 G. C. Greubel, Table of n, a(n) for n = 0..5000 %H A263571 Michael Somos, Introduction to Ramanujan theta functions, 2019. %H A263571 Eric Weisstein's World of Mathematics, Ramanujan Theta Functions. %F A263571 Expansion of chi(x) * phi(x^2) * psi(-x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions. %F A263571 Expansion of q^(-1/3) * eta(q^3) * eta(q^4)^4 * eta(q^12) / (eta(q) * eta(q^6) * eta(q^8)^2) in powers of q. %F A263571 a(n) = b(3*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = (-1)^e, b(p^e) = e+1 if p == 1, 7 (mod 24), b(p^e) = (e+1) * (-1)^e if p == 5, 11 (mod 24), b(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24). %F A263571 G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 24^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263577. %F A263571 a(n) = A115660(3*n + 1) = A192013(3*n + 1) = A128581(6*n + 2). %F A263571 a(2*n) = A261115(n). a(2*n + 1) = A263548(n). a(4*n + 1) = a(n). a(4*n + 3) = 2 * A128582(n). %F A263571 a(8*n + 4) = a(8*n + 6) = 0. a(8*n) = A113780(n). a(8*n + 2) = 2 * A260089(n). %F A263571 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(6) = 1.282549... . - _Amiram Eldar_, Dec 28 2023 %e A263571 G.f. = 1 + x + 2*x^2 + 2*x^3 + x^5 + 2*x^7 + 3*x^8 + 2*x^9 + 2*x^10 + ... %e A263571 G.f. = q + q^4 + 2*q^7 + 2*q^10 + q^16 + 2*q^22 + 3*q^25 + 2*q^28 + ... %t A263571 a[ n_] := If[ n < 0, 0, With[ {m = 3 n + 1}, Sum[ KroneckerSymbol[ 2, d] KroneckerSymbol[ -3, m/d], {d, Divisors[ m]}]]]; %t A263571 a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] EllipticTheta[ 3, 0, x^2] EllipticTheta[ 2, Pi/4, x^(3/2)] / (2^(1/2) x^(3/8)), {x, 0, n}]; %o A263571 (PARI) {a(n) = my(m); if( n<0, 0, m = 3*n + 1; sumdiv( m, d, kronecker( 2, d) * kronecker( -3, m/d)))}; %o A263571 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^4 + A)^4 * eta(x^12 + A) / (eta(x + A) * eta(x^8 + A)^2), n))}; %Y A263571 Cf. A113780, A115660, A128581, A128582, A192013, A260089, A261115, A263548, A263577. %Y A263571 Cf. A000122, A000700, A010054, A121373. %K A263571 nonn,easy %O A263571 0,3 %A A263571 _Michael Somos_, Oct 21 2015 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE