A271593 - OEIS

%I #19 Feb 16 2025 08:33:33

%S 1,1,2,2,4,5,8,10,15,18,26,32,44,54,72,88,115,140,180,218,276,333,416,

%T 500,618,740,906,1080,1312,1558,1880,2224,2666,3143,3746,4402,5220,

%U 6114,7216,8426,9903,11530,13498,15672,18280,21168,24608,28424,32940,37956

%N Expansion of psi(-x^3) / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H Vaclav Kotesovec, <a href="/A271593/b271593.txt">Table of n, a(n) for n = 0..2000</a>

%H Andrew Sills, <a href="https://works.bepress.com/andrew_sills/40/">Rademacher-Type Formulas for Restricted Partition and Overpartition Functions</a>, Ramanujan Journal, 23 (1-3): 253-264, 2010.

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of f(x, x^5) / phi(-x^2) in powers of x where phi(), f(, ) are Ramanujan theta functions.

%F Expansion of q^(-1/3) * eta(q^3) * eta(q^12) / (eta(q) * eta(q^6)) in powers of q.

%F Euler transform of period 12 sequence [ 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, ...].

%F G.f.: Sum_{k>=0} x^(k*(k+1)) * (-x^2, x^2)_k / (x, x)_{2*k+1}.

%F a(n) ~ Pi * BesselI(1, Pi*sqrt(3*n+1) / sqrt(6)) / (4*sqrt(3*n+1)) ~ exp(sqrt(n/2)*Pi) / (2^(9/4)*sqrt(3)*n^(3/4)) * (1 + (Pi/6 - 3/(4*Pi))/sqrt(2*n) + (Pi^2/144 - 15/(64*Pi^2) - 5/16)/n). - _Vaclav Kotesovec_, Apr 18 2016, extended Jan 10 2017

%e G.f. = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 5*x^5 + 8*x^6 + 10*x^7 + 15*x^8 + ...

%e G.f. = q + q^4 + 2*q^7 + 2*q^10 + 4*q^13 + 5*q^16 + 8*q^19 + 10*q^22 + ...

%t nmax = 50; CoefficientList[Series[Product[(1-x^(3*k)) * (1+x^(6*k)) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Apr 18 2016 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^6 + A)), n))};

%K nonn,changed

%O 0,3

%A _Michael Somos_, Apr 10 2016