A262950 - OEIS

%I #24 Mar 12 2021 22:24:48

%S 1,1,1,1,1,2,2,3,3,3,4,5,6,6,7,8,9,11,12,14,15,17,20,22,25,28,31,34,

%T 38,43,48,53,58,64,71,79,88,96,105,115,127,140,154,168,183,200,219,

%U 240,262,285,310,337,367,400,435,472,511,554,601,652,707,764,825

%N Expansion of q * chi(-q^3) * chi(-q^13) / (chi(-q) * chi(-q^39)) in powers of q where chi() is a Ramanujan theta function.

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

%H G. C. Greubel, <a href="/A262950/b262950.txt">Table of n, a(n) for n = 1..1000</a>

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

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

%F Expansion of eta(q^2) * eta(q^3) * eta(q^13) * eta(q^78) / (eta(q) * eta(q^6) * eta(q^26) * eta(q^39)) in powers of q.

%F Euler transform of a period 78 sequence.

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u^2 - v) * (v - w^2) - 2 * u*w * (1 + v)^2.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (78 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128519.

%F G.f.: x * Product_{k>0} P(x^k) where P(x) is the 78th cyclotomic polynomial of degree 24.

%F Convolution inverse of A128519.

%F a(n) ~ exp(2*Pi*sqrt(2*n/39)) / (2^(3/4) * 39^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Nov 29 2019

%e G.f. = q + q^2 + q^3 + q^4 + q^5 + 2*q^6 + 2*q^7 + 3*q^8 + 3*q^9 + 3*q^10 + ...

%t eta[q_] := q^(1/24)*QPochhammer[q]; b := eta[q^2]*eta[q^3]*eta[q^13]* eta[q^78]/(eta[q]*eta[q^6]*eta[q^26]*eta[q^39]); a:= CoefficientList[ Series[q*b, {q, 0, 80}], q]; Drop[Table[a[[n]], {n, 1, 80}], 2] (* _G. C. Greubel_, Jul 03 2018 *)

%t nmax = 100; CoefficientList[Series[Product[(1 + x^k) * (1 + x^(39*k)) / ((1 + x^(3*k)) * (1 + x^(13*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 29 2019 *)

%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^13 + A) * eta(x^78 + A) / (eta(x + A) * eta(x^6 + A) * eta(x^26 + A) * eta(x^39 + A)), n))};

%Y Cf. A128519.

%K nonn

%O 1,6

%A _Michael Somos_, Nov 19 2015