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A320050 Expansion of (psi(x) / phi(x))^7 in powers of x where phi(), psi() are Ramanujan theta functions.

%I A320050 #20 Feb 16 2025 08:33:56
%S A320050 1,-7,35,-140,483,-1498,4277,-11425,28889,-69734,161735,-362271,
%T A320050 786877,-1662927,3428770,-6913760,13660346,-26492361,50504755,
%U A320050 -94766875,175221109,-319564227,575387295,-1023624280,1800577849,-3133695747,5399228149,-9214458260,15584195428
%N A320050 Expansion of (psi(x) / phi(x))^7 in powers of x where phi(), psi() are Ramanujan theta functions.
%C A320050 In general, for b > 0 and (psi(x) / phi(x))^b, a(n) ~ (-1)^n * b^(1/4) * exp(Pi*sqrt(b*(n/2))) / (2^(b + 7/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 06 2018
%H A320050 Seiichi Manyama, <a href="/A320050/b320050.txt">Table of n, a(n) for n = 0..10000</a>
%H A320050 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A320050 Convolution inverse of A029844.
%F A320050 Expansion of q^(-7/8) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^7 in powers of q.
%F A320050 a(n) ~ (-1)^n * 7^(1/4) * exp(Pi*sqrt((7*n)/2)) / (256*2^(3/4)*n^(3/4)). - _Vaclav Kotesovec_, Oct 06 2018
%t A320050 nmax = 50; CoefficientList[Series[Product[((1-x^k) * (1-x^(4*k))^2 / (1-x^(2*k))^3)^7, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 06 2018 *)
%Y A320050 (psi(x) / phi(x))^b: A083365 (b=1), A079006 (b=2), A187053 (b=3), A001938 (b=4), A195861 (b=5), A320049 (b=6), this sequence (b=7).
%Y A320050 Cf. A029844.
%K A320050 sign
%O A320050 0,2
%A A320050 _Seiichi Manyama_, Oct 04 2018