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%I #19 Mar 12 2021 22:24:42
%S 1,4,10,24,51,104,206,384,697,1228,2112,3568,5898,9592,15358,24256,
%T 37850,58340,88980,134344,200972,298112,438538,640256,928041,1336104,
%U 1911436,2717776,3842110,5401784,7555012,10514176,14562432,20077672
%N Expansion of q * (chi(-q) * chi(-q^5))^-4 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="/A093831/b093831.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 Expansion of (eta(q^2) * eta(q^10) / (eta(q) * eta(q^5)))^4 in powers of q.
%F Euler transform of period 10 sequence [ 4, 0, 4, 0, 8, 0, 4, 0, 4, 0, ...].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v*(1 + 8*u + 16*u*v).
%F G.f.: x * (Product_{k>0} (1 - x^(10*k - 5)) * (1 - x^(2*k - 1)))^-4.
%F Convolution inverse of A132040. - _Michael Somos_, Apr 26 2015
%F a(n) ~ exp(2*Pi*sqrt(2*n/5)) / (16 * 2^(3/4) * 5^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2015
%e G.f. = q + 4*q^2 + 10*q^3 + 24*q^4 + 51*q^5 + 104*q^6 + 206*q^7 + 384*q^8 + ...
%t a[ n_] := SeriesCoefficient[ q (QPochhammer[ -q, q] QPochhammer[ -q^5, q^5] )^4, {q, 0, n}]; (* _Michael Somos_, Apr 26 2015 *)
%t nmax = 40; Rest[CoefficientList[Series[x * Product[1/((1 - x^(10*k - 5)) * (1 - x^(2*k - 1)))^4, {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Sep 07 2015 *)
%o (PARI) {a(n) = if( n<1, 0, n--; polcoeff( (1 / prod(k=1, (n+5)\10, 1 - x^(10*k - 5), 1 + x * O(x^n)) / prod(k=1, (n+1)\2, 1 - x^(2*k - 1), 1 + x * O(x^n)))^4, n))};
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^10 + A) / (eta(x + A) * eta(x^5 + A)))^4, n))};
%Y Cf. A132040.
%K nonn
%O 1,2
%A _Michael Somos_, Apr 17 2004, Oct 04 2004
%E Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, Jun 05 2007
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