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A127692 Expansion of psi(x^4) + x * psi(x^12) in powers of x where psi() is a Ramanujan theta function.

%I A127692 #30 Aug 30 2025 17:16:01
%S A127692 1,1,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,
%T A127692 0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,
%U A127692 0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A127692 Expansion of psi(x^4) + x * psi(x^12) in powers of x where psi() is a Ramanujan theta function.
%C A127692 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A127692 a(n) = 1 if n is four times a triangular number or one more than twelve times a triangular number else 0. - _Michael Somos_, Jul 19 2012
%H A127692 Antti Karttunen, <a href="/A127692/b127692.txt">Table of n, a(n) for n = 0..10000</a>
%H A127692 Richard Blecksmith, John Brillhart, and Irving Gerst, <a href="http://dx.doi.org/10.1090/S0025-5718-1988-0942157-2">Some infinite product identities</a>, Math. Comp. 51 (1988), no. 183, 301-314. MR0942157 (89f:05017)
%H A127692 Shaun Cooper and Michael Hirschhorn, <a href="http://dx.doi.org/10.1216/rmjm/1008959672">On some infinite product identities</a>, Rocky Mountain J. Math., 31 (2001), 131-139. see p. 134 Theorem 5.
%H A127692 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A127692 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A127692 Euler transform of period 24 sequence [ 1, -1, 0, 1, -1, 1, -1, 0, 0, 0, 1, -1, 1, 0, 0, 0, -1, 1, -1, 1, 0, -1, 1, -1, ...].
%F A127692 a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = 1, else b(p^e) = (1 + (-1)^e)/2.
%F A127692 a(3*n + 1) = a(n), a(3*n + 2) = a(4*n + 2) = a(4*n + 3) = a(6*n + 3) = 0.
%F A127692 a(2*n) = A005369(n). a(4*n) = A010054(n). a(6*n) = A089806(n). a(12*n) = A080995(n).
%F A127692 G.f.: Sum_{k>0} x^(2k(k-1)) +x^(6k(k-1)+1) = Product_{k>0} (1-x^(24k)) (1-x^(24k-5)) (1-x^(24k-7)) (1-x^(24k-17)) (1-x^(24k-19)) (1+x^(12k-1)) (1+x^(12k-4)) (1+x^(12k-6)) (1+x^(12k-8)) (1+x^(12k-11)).
%F A127692 From _Michael Somos_, Jul 19 2012: (Start)
%F A127692 Expansion of f(x, -x^5) * f(-x^4, -x^8) / f(x, -x) in powers of x where f(,) is the Ramanujan two-variable theta function.
%F A127692 G.f.: (Sum_{k in Z} x^(2*k*(k + 1)) + x^(6*k*(k + 1) + 1)) / 2.
%F A127692 a(n) = A195198(2*n + 1). (End)
%F A127692 Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = 1/sqrt(2) + 1/sqrt(6) = 1.115355... (A145439). - _Amiram Eldar_, Dec 29 2023
%e A127692 G.f. = 1 + x + x^4 + x^12 + x^13 + x^24 + x^37 + x^40 + x^60 + x^73 + x^84 + ...
%e A127692 G.f. = q + q^3 + q^9 + q^25 + q^27 + q^49 + q^75 + q^81 + q^121 + q^147 + q^169 + ...
%o A127692 (PARI) {a(n) = issquare(2*n + 1) + issquare(6*n + 3)};
%o A127692 (PARI) {a(n) = n = 2*n + 1; issquare(n) || issquare(3*n)};
%Y A127692 Cf. A005369, A010054, A080995, A089806, A145439.
%Y A127692 Cf. A000122, A000700, A010054, A121373.
%K A127692 nonn,easy,changed
%O A127692 0,1
%A A127692 _Michael Somos_, Jan 19 2007