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

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, 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, 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

Offset: 0

Author

Michael Somos, Oct 28 2005

Keywords

• nonn

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

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

Examples

			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 + ...
G.f. = q + q^9 + 2*q^17 + 3*q^25 + 2*q^41 + q^49 + 4*q^65 + 2*q^73 + q^81 + ...
		

References

• B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 114 Entry 8(vi).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A008441, A053692.

Programs

Mathematica

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 *)

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))};

Formula

Expansion of chi(x) * f(x)^2 in powers of x where chi(), f() are Ramanujan theta functions. - Michael Somos, Jul 24 2012

Expansion of q^(-1/8) * eta(q^4)^5 / (eta(q) * eta(q^8)^2) in powers of q.

Euler transform of period 8 sequence [ 1, 1, 1, -4, 1, 1, 1, -2, ...].

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

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).

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