|
| |
|
|
A033761
|
|
Product t2(q^d); d | 2, where t2 = theta2(q)/(2*q^(1/4)).
|
|
5
| |
|
|
1, 1, 1, 2, 0, 1, 2, 1, 1, 1, 1, 0, 3, 1, 0, 2, 1, 1, 1, 0, 1, 3, 1, 2, 0, 0, 1, 2, 1, 0, 3, 1, 0, 2, 1, 1, 2, 0, 1, 0, 2, 1, 2, 1, 0, 3, 0, 1, 3, 0, 0, 2, 1, 0, 0, 1, 2, 4, 1, 1, 0, 1, 1, 1, 0, 1, 3, 1, 1, 0, 1, 1, 2, 1, 0, 3, 0, 1, 4, 0, 1, 0, 1, 0, 2, 1, 1, 2, 0, 0, 2, 2, 1, 3, 0, 0, 2, 2, 1, 0, 2, 1, 0, 1, 0
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,4
|
|
|
COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(
q) (A010054), chi(q) (A000700).
Also the number of representations of n as the sum of a triangular number and twice a triangular number. - James A. Sellers (sellersj(AT)math.psu.edu), Dec 21 2005
|
|
|
LINKS
| M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
|
|
|
FORMULA
| Euler transform of period 4 sequence [1, 0, 1, -2, ...]. - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 14 2004
Expansion of psi(q) * psi(q^2) in powers of q where psi() is a Ramanujan theta function.
Expansion of q^(-3/8) * eta(q^2) * eta^2(q^4) / eta(q) in powers of q. - Michael Somos, Jul 05 2006
Expansion of q^(-3/4) * (theta_2(q) * theta_2(q^2)) / 4 in powers of q^2. - Michael Somos, Jul 05 2006
Given g.f. A(x), then B(x) = x^3 * A(x^8) satisfies 0 = f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6) = u1^4*u6^2 + 3*u2^2*u3^4 - 4*u1*u2*u3*u6 * (u2^2 + 3*u6^2) - Michael Somos, Jul 05 2006
a(n) = A002325(8*n+3)/2. [Hirschhorn] - R. J. Mathar, Mar 23 2011
a(n) = A027414(8*n + 3). - Michael Somos, Nov 16 2011
|
|
|
EXAMPLE
| 1 + x + x^2 + 2*x^3 + x^5 + 2*x^6 + x^7 + x^8 + x^9 + x^10 + 3*x^12 + ...
q^3 + q^11 + q^19 + 2*q^27 + q^43 + 2*q^51 + q^59 + q^67 + q^75 + q^83 + ...
|
|
|
MAPLE
| sigmamr := proc(n, m, r) local a, d ; a := 0 ; for d in numtheory[divisors](n) do if modp(d, m) = r then a := a+1 ; end if; end do: a; end proc:
A002325 := proc(n) sigmamr(n, 8, 1)+sigmamr(n, 8, 3)-sigmamr(n, 8, 5)-sigmamr(n, 8, 7) ; end proc:
A033761 := proc(n) A002325(8*n+3)/2 ; end proc:
seq(A033761(n), n=0..90) ; # R. J. Mathar, Mar 23 2011
|
|
|
MATHEMATICA
| a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^2] / 4, {q, 0, 2 n + 3/4}] (* Michael Somos, Nov 16 2011 *)
|
|
|
PROG
| (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^2 / eta(x + A), n))} /* Michael Somos, Jul 05 2006 */
|
|
|
CROSSREFS
| Cf. A027414, A097723.
Sequence in context: A085097 A117997 A079684 * A033805 A033797 A033793
Adjacent sequences: A033758 A033759 A033760 * A033762 A033763 A033764
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 14 2004
|
| |
|
|
