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

0,3

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

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

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

Example

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 + ...

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 *)

Prog

(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))};

Crossrefs

Cf. A008441, A053692.

Sequence in context: A126832 A068908 A226192 * A039703 A171935 A103180

Adjacent sequences: A113404 A113405 A113406 * A113408 A113409 A113410

Keyword

nonn

Author

Michael Somos, Oct 28 2005

Status

approved