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%I A187154 #40 Feb 16 2025 08:33:14
%S A187154 1,2,4,8,15,26,44,72,114,178,272,408,605,884,1276,1824,2580,3616,5028,
%T A187154 6936,9498,12922,17468,23472,31369,41700,55156,72616,95172,124202,
%U A187154 161436,209016,269616,346562,443952,566856,721530,915642,1158608,1461968,1839789
%N A187154 Expansion of psi(x^4) / phi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.
%C A187154 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A187154 Since phi(-x) = 1 + 2*Sum_{k >= 1} (-1)^k*x^(k^2) == 1 (mod 2), it follows that the g.f. psi(x^4) / phi(-x) == psi(x^4) == Sum_{k >= 0} x^(2*k*(k+1)) (mod 2). Hence a(n) is odd iff n = 2*k*(k + 1) for some nonegative integer k. - _Peter Bala_, Jan 07 2025
%H A187154 G. C. Greubel, <a href="/A187154/b187154.txt">Table of n, a(n) for n = 0..1000</a>
%H A187154 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A187154 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A187154 Expansion of q^(-1/2) * eta(q^2) * eta(q^8)^2 / (eta(q)^2 * eta(q^4)) in powers of q.
%F A187154 Euler transform of period 8 sequence [ 2, 1, 2, 2, 2, 1, 2, 0, ...].
%F A187154 G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32^(-1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A093085.
%F A187154 Convolution inverse of A093085. Convolution square is A107035.
%F A187154 a(n) ~ exp(sqrt(n)*Pi)/(16*n^(3/4)). - _Vaclav Kotesovec_, Sep 10 2015
%e A187154 1 + 2*x + 4*x^2 + 8*x^3 + 15*x^4 + 26*x^5 + 44*x^6 + 72*x^7 + 114*x^8 + ...
%e A187154 q + 2*q^3 + 4*q^5 + 8*q^7 + 15*q^9 + 26*q^11 + 44*q^13 + 72*q^15 + 114*q^17 + ...
%t A187154 nmax = 50; CoefficientList[Series[Product[(1 + x^k)^2 * (1 + x^(2*k)) * (1 + x^(4*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 10 2015 *)
%t A187154 a[n_]:= SeriesCoefficient[EllipticTheta[2, 0, q^2]/(2*Sqrt[q]* EllipticTheta[3, 0, -q]), {q, 0, n}]; Table[A187154[n], {n, 0, 50}] (* _G. C. Greubel_, Dec 04 2017 *)
%o A187154 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)), n))}
%Y A187154 Cf. A001935, A001936, A0022568, A093085, A107035.
%K A187154 nonn,easy,changed
%O A187154 0,2
%A A187154 _Michael Somos_, Mar 08 2011

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