A168370 a(n) = coefficient of x^n in the (3^n)-th power of 1 + Sum_{k>=0} x^(3^k) for n>=0.

1, 3, 36, 2952, 1670220, 6781419711, 204255279577440, 47027922196061266047, 84798672814179921118709052, 1219732878003607687535196405346440, 141916059665284234793191571472586402539060

Offset

0,2

Links

Table of n, a(n) for n=0..10.

Formula

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

Example

G.f.: A(x) = 1 + 3*x + 36*x^2 + 2952*x^3 + 1670220*x^4 +...
Let F(x) = 1 + x + x^3 + x^9 + x^27 + x^81 +...+ x^(3^n) +...
then A(x) = 1 + log(F(3x)) + log(F(9x))^2/2! + log(F(27x))^3/3! +...+ log(F(3^n*x))^n/n! +...
Also, coefficients in powers F(x)^(3^n) begin:
F^1:[(1),1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,...];
F^3:[1,(3),3,4,6,3,3,3,0,4,6,3,6,6,0,3,0,0,3,3,0,3,0,0,0,0,0,4,...];
F^9:[1,9,(36),93,198,378,624,918,1269,1606,1908,2277,2634,3060,...];
F^27:[1,27,351,(2952),18252,89505,366561,1300455,4101435,11713287,...];
F^81:[1,81,3240,85401,(1670220),25877556,331198416,3605580540,...];
F^243:[1,243,29403,2362284,141781266,(6781419711),269282151567,...];
F^729:[1,729,265356,64305333,11671816338,1692529329582,(204255279577440), ...]; ...
where the coefficients in parenthesis form the initial terms of this sequence.

Prog

(PARI) {a(n)=local(G=1+sum(m=0, ceil(log(n+3)/log(3)), x^(3^m))+x*O(x^n)); polcoeff(G^(3^n), n)}
(PARI) {a(n)=local(G=1+sum(m=0, ceil(log(n+3)/log(3)), x^(3^m))+x*O(x^n)); polcoeff(sum(m=0, n, log(subst(G, x, 3^m*x))^m/m!), n)}

Crossrefs

Cf. A168369 (variant).

Sequence in context: A102579 A368076 A136393 * A325907 A158093 A163966

Adjacent sequences: A168367 A168368 A168369 * A168371 A168372 A168373

Keyword

nonn

Author

Paul D. Hanna, Nov 24 2009

Status

approved