A166953 Number of ways of writing n as the sum of 3^n squares.

1, 6, 144, 23400, 26620002, 216778910040, 13069351570163616, 6019308484501930839936, 21708290476794620365667887680, 624502420526473667139055032092300382

Offset

0,2

Links

Table of n, a(n) for n=0..9.

Formula

a(n) equals the coefficient of x^n in the (3^n)-th power of Jacobi theta_3(x).

G.f.: A(x) = Sum_{n>=0} log( theta_3(3^n*x) )^n/n! where theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).

Example

G.f.: A(x) = 1 + 6*x + 144*x^2 + 23400*x^3 + 26620002*x^4 +...
Let F(x) = theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2),
then A(x) = 1 + log(F(3*x)) + log(F(9*x))^2/2! + log(F(27*x))^3/3! + log(F(81*x))^4/4! + ...
Illustrate a(n) = [x^n] F(x)^(3^n) by forming a table of
coefficients in powers F(x)^(3^n), which begin:
F^(3^0): [(1), 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, ...];
F^(3^1): [1, (6), 12, 8, 6, 24, 24, 0, 12, 30, 24, 24, 8, 24, ...];
F^(3^2): [1, 18, (144), 672, 2034, 4320, 7392, 12672, 22608, ...];
F^(3^3): [1, 54, 1404, (23400), 280854, 2586168, 19014840, ...];
F^(3^4): [1, 162, 12960, 682560, (26620002), 819916992, ...];
F^(3^5): [1, 486, 117612, 18896328, 2267559846, (216778910040), ...]; ...
and noting that the coefficients along the diagonal (in parenthesis)
form the initial terms of this sequence.

Prog

(PARI) {a(n)=local(THETA3=1+2*sum(k=1, sqrtint(n), x^(k^2))+x*O(x^n)); polcoeff(THETA3^(3^n), n)}
(PARI) {a(n)=local(THETA3=1+2*sum(k=1, sqrtint(n), x^(k^2))+x*O(x^n)); polcoeff(sum(k=0, n, log(subst(THETA3, x, 3^k*x))^k/k!), n)}

Crossrefs

Cf. A000122, A005875, A008452, variants: A166947, A158113.

Sequence in context: A090443 A307416 A133460 * A280847 A041271 A196964

Adjacent sequences: A166950 A166951 A166952 * A166954 A166955 A166956

Keyword

nonn

Author

Paul D. Hanna, Oct 26 2009

Status

approved