A033791 Product t2(q^d); d | 32, where t2 = theta2(q)/(2*q^(1/4)).

1, 1, 1, 2, 1, 2, 3, 3, 2, 3, 4, 3, 6, 5, 5, 7, 6, 6, 7, 6, 9, 10, 8, 9, 12, 12, 11, 16, 12, 14, 18, 16, 15, 19, 19, 19, 21, 18, 20, 22, 26, 24, 28, 27, 23, 32, 30, 29, 35, 33, 35, 40, 38, 36, 44, 45, 42, 49, 47, 44, 58, 54, 50, 60, 56, 56, 62, 58, 64, 73, 66, 66

Offset

0,4

Comments

theta_2 is the standard Jacobi theta-constant function, Sum_{m=-oo..oo} q^((m+1/2)^2) = 2*q^(1/4)*(1+q^2+q^6+q^12+q^20+q^30+...) - see A098108.

Thus the above function t2 equals t2(z) = 1 + z^2 + z^6 + z^12 + ... = sum_{n>=0} z^(n(n+1)) = sum_{n>=0} z^A002378(n). But note that the sequence lists only the nonzero coefficients which are those of even powers. In other words, this is the sequence of coefficients of product(t'(x^2^d), d=0..5) with t'(z) = 1 + z + z^3 + z^6 + ... = sum_{n>=0} z^(n(n+1)/2) = sum_{n>=0} z^A000217(n). - M. F. Hasler, Oct 15 2014

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Example

Here is the beginning of the theta-series theta_2:
2*q^(1/4) + 2*q^(9/4) + 2*q^(25/4) + 2*q^(49/4) + 2*q^(81/4) + 2*q^(121/4) + 2*q^(169/4) + 2*q^(225/4) + 2*q^(289/4) + 2*q^(361/4) + 2*q^(441/4) + 2*q^(529/4) + 2*q^(625/4) + 2*q^(729/4) + ...
t(x)*t(x^2)*t(x^4)*t(x^8)*t(x^16)*t(x^32) = 1 + x + x^2 + 2*x^3 + x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 2*x^8 + 3*x^9 + 4*x^10 + 3*x^11 + .... - M. F. Hasler, Oct 15 2014

Prog

(PARI) my(x='x+O('x^99), t(z)=sum(i=0, 10, z^((i+1)*i/2))); Vec(prod(d=0, 5, t(x^2^d))) \\ M. F. Hasler, Oct 15 2014

Crossrefs

Cf. A098108.

Sequence in context: A054242 A033767 A033775 * A039913 A108617 A092683

Adjacent sequences: A033788 A033789 A033790 * A033792 A033793 A033794

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Seiichi Manyama, May 24 2017

Status

approved