This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A138506 #18 Feb 16 2025 08:33:08
%S A138506 1,5,5,-10,-15,5,-10,-30,25,35,5,60,30,-60,-30,-10,-55,-80,35,100,-15,
%T A138506 60,60,-110,-50,5,-60,-100,90,150,-10,160,105,-120,-80,-30,-105,-180,
%U A138506 100,120,25,210,60,-210,-180,35,-110,-230,110,215,5,160,180,-260,-100
%N A138506 Expansion of f(q)^5 / f(q^5) in powers of q where f() is a Ramanujan theta function.
%C A138506 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A138506 G. C. Greubel, <a href="/A138506/b138506.txt">Table of n, a(n) for n = 0..10000</a>
%H A138506 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A138506 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A138506 Expansion of eta(q^2)^15 * eta(q^5) * eta(q^20) / (eta(q)^5 * eta(q^4)^5 * eta(q^10)^3) in powers of q.
%F A138506 Euler transform of period 20 sequence [ 5, -10, 5, -5, 4, -10, 5, -5, 5, -8, 5, -5, 5, -10, 4, -5, 5, -10, 5, -4, ...].
%F A138506 a(n) = 5*b(n) where b() is multiplicative with b(2^e) = ((-2)^(n+1) - 1) / 3, b(p^e) = ((-p)^(e+1) - 1) / (-p - 1) if p == 2, 3 (mod 5), b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 4 (mod 5).
%F A138506 G.f.: Product_{k>0} (1 - (-x)^k)^5 / (1 - (-x)^(5*k)).
%F A138506 a(n) = (-1)^n * A109064(n). a(n) = 5 * A138507(n) unless n=0.
%F A138506 Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(3*sqrt(5)) = 1.471273... . - _Amiram Eldar_, Jan 29 2024
%e A138506 G.f. = 1 + 5*q + 5*q^2 - 10*q^3 - 15*q^4 + 5*q^5 - 10*q^6 - 30*q^7 + 25*q^8 + ...
%t A138506 a[ n_] := SeriesCoefficient[ QPochhammer[ -q]^5 / QPochhammer[ -q^5], {q, 0, n}]; (* _Michael Somos_, May 24 2015 *)
%t A138506 a[ n_] := If[ n < 1, Boole[n == 0], -5 (-1)^n DivisorSum[ n, # KroneckerSymbol[ 5, #] &]]; (* _Michael Somos_, May 24 2015 *)
%o A138506 (PARI) {a(n) = if( n<1, n==0, -5 * (-1)^n * sumdiv(n, d, d * kronecker(5, d)))};
%o A138506 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(-x + A)^5 / eta(-x^5 + A), n))};
%Y A138506 Cf. A109064, A138507.
%Y A138506 Cf. A000122, A000700, A010054, A121373.
%K A138506 sign
%O A138506 0,2
%A A138506 _Michael Somos_, Mar 21 2008, Mar 25 2008