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A113407 Expansion of psi(x) * phi(x^2) in powers of x where psi(), phi() are Ramanujan theta functions.

%I A113407 #26 Feb 16 2025 08:32:59
%S A113407 1,1,2,3,0,2,1,0,4,2,1,2,2,0,2,1,0,2,4,2,0,3,0,4,2,0,0,0,3,2,2,0,2,4,
%T A113407 0,2,3,0,4,2,0,0,2,0,2,1,2,4,0,0,2,2,0,6,2,1,2,2,0,0,4,0,0,4,0,2,1,0,
%U A113407 4,0,0,2,2,4,2,2,0,2,5,0,2,0,2,0,2,0,4,4,0,0,0,1,0,4,0,2,2,0,4,4,2,2,0,0,2
%N A113407 Expansion of psi(x) * phi(x^2) in powers of x where psi(), phi() are Ramanujan theta functions.
%C A113407 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A113407 Bisection of A008441. Number of ways to write n as two times a square plus a triangular number [Hirschhorn]. - _R. J. Mathar_, Mar 23 2011
%D A113407 B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 114 Entry 8(vi).
%H A113407 G. C. Greubel, <a href="/A113407/b113407.txt">Table of n, a(n) for n = 0..5000</a>
%H A113407 M. D. Hirschhorn, <a href="http://dx.doi.org/10.1016/j.disc.2004.08.045">The number of representations of a number by various forms</a>, Discrete Mathematics 298 (2005), 205-211
%H A113407 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A113407 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A113407 Expansion of chi(x) * f(x)^2 in powers of x where chi(), f() are Ramanujan theta functions. - _Michael Somos_, Jul 24 2012
%F A113407 Expansion of q^(-1/8) * eta(q^4)^5 / (eta(q) * eta(q^8)^2) in powers of q.
%F A113407 Euler transform of period 8 sequence [ 1, 1, 1, -4, 1, 1, 1, -2, ...].
%F A113407 a(n) = b(8*n + 1), where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 +(-1)^e) / 2 if p == 3 (mod 4), b(p^e) = e+1 if p == 1 (mod 4). - _Michael Somos_, Jul 24 2012
%F A113407 G.f.: (Sum_{k in Z} x^(2*k^2)) * (Sum_{k>=0} x^((k^2 + k)/2)) = Sum_{k>=0} (-1)^k * (x^(2*k + 1) + 1) / (x^(2*k + 1) - 1) * x^((k^2 + k)/2).
%F A113407 a(9*n + 4) = a(9*n + 7) = 0. a(9*n + 1) = a(n). a(n) = A008441(2*n). - _Michael Somos_, Jul 24 2012
%e A113407 G.f. = 1 + x + 2*x^2 + 3*x^3 + 2*x^5 + x^6 + 4*x^8 + 2*x^9 + x^10 + 2*x^11 + ...
%e A113407 G.f. = q + q^9 + 2*q^17 + 3*q^25 + 2*q^41 + q^49 + 4*q^65 + 2*q^73 + q^81 + ...
%t A113407 phi[x_] := EllipticTheta[3, 0, x]; psi[x_] := (1/2)*x^(-1/8)*EllipticTheta[2, 0, x^(1/2)]; s = Series[psi[x]*phi[x^2], {x, 0, 104}]; a[n_] := Coefficient[s, x, n] ; Table[a[n], {n, 0, 104}] (* _Jean-François Alcover_, Feb 17 2015 *)
%o A113407 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^5 / (eta(x + A) * eta(x^8 + A)^2), n))};
%Y A113407 Cf. A008441, A053692.
%K A113407 nonn
%O A113407 0,3
%A A113407 _Michael Somos_, Oct 28 2005