A166953 - OEIS

%I #2 Mar 30 2012 18:37:18

%S 1,6,144,23400,26620002,216778910040,13069351570163616,

%T 6019308484501930839936,21708290476794620365667887680,

%U 624502420526473667139055032092300382

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

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

%F 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).

%e G.f.: A(x) = 1 + 6*x + 144*x^2 + 23400*x^3 + 26620002*x^4 +...

%e Let F(x) = theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2),

%e 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! + ...

%e Illustrate a(n) = [x^n] F(x)^(3^n) by forming a table of

%e coefficients in powers F(x)^(3^n), which begin:

%e F^(3^0): [(1), 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, ...];

%e F^(3^1): [1, (6), 12, 8, 6, 24, 24, 0, 12, 30, 24, 24, 8, 24, ...];

%e F^(3^2): [1, 18, (144), 672, 2034, 4320, 7392, 12672, 22608, ...];

%e F^(3^3): [1, 54, 1404, (23400), 280854, 2586168, 19014840, ...];

%e F^(3^4): [1, 162, 12960, 682560, (26620002), 819916992, ...];

%e F^(3^5): [1, 486, 117612, 18896328, 2267559846, (216778910040), ...]; ...

%e and noting that the coefficients along the diagonal (in parenthesis)

%e form the initial terms of this sequence.

%o (PARI) {a(n)=local(THETA3=1+2*sum(k=1,sqrtint(n),x^(k^2))+x*O(x^n));polcoeff(THETA3^(3^n),n)}

%o (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)}

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

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 26 2009