A123632 - OEIS

%I #19 Mar 12 2021 22:24:44

%S 1,1,1,3,3,5,8,9,13,19,24,31,42,52,67,91,110,137,180,217,272,344,412,

%T 509,633,762,925,1132,1354,1631,1984,2353,2808,3382,3992,4747,5658,

%U 6644,7850,9291,10882,12772,15016,17512,20455,23944,27796,32311,37633,43529

%N Expansion of q / (chi(-q) * chi(-q^3) * chi(-q^5) * chi(-q^15)) 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 Seiichi Manyama, <a href="/A123632/b123632.txt">Table of n, a(n) for n = 1..10000</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 Euler transform of period 30 sequence [ 1, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 4, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 0, ...].

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

%F Expansion of (eta(q^2) * eta(q^6) * eta(q^10) * eta(q^30)) / (eta(q) * eta(q^3) * eta(q^5) * eta(q^15)) in powers of q.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132321.

%F G.f.: x * Product_{k>0} (1 + x^k) * (1 + x^(3*k)) * (1 + x^(5*k)) * (1 + x^(15*k)).

%F Convolution inverse of A132321.

%F a(n) ~ exp(2*Pi*sqrt(2*n/15)) / (2^(11/4) * 15^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2015

%F G.f.: r * s = (r + s - 1)/2 where r = x * Product_{k>0} (1 - x^(2*k)) * (1 - x^(30*k)) / ((1 - x^(3*k)) * (1 - x^(5*k))), s = Product_{k>0} (1 - x^(6*k)) * (1 - x^(10*k)) / ((1 - x^k) * (1 - x^(15*k))). - _Seiichi Manyama_, May 05 2017

%e G.f. = q + q^2 + q^3 + 3*q^4 + 3*q^5 + 5*q^6 + 8*q^7 + 9*q^8 + 13*q^9 + 19*q^10 + ...

%t nmax = 40; Rest[CoefficientList[Series[x * Product[(1 + x^k) * (1 + x^(3*k)) * (1 + x^(5*k)) * (1 + x^(15*k)), {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Sep 07 2015 *)

%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A) * eta(x^10 + A) * eta(x^30 + A) / (eta(x + A) * eta(x^3 + A) * eta(x^5 + A) * eta(x^15 + A)), n))};

%Y Cf. A094022 (r), A094023 (s), A132321.

%K nonn

%O 1,4

%A _Michael Somos_, Oct 03 2006, Jan 12 2009