A168369 a(n) = coefficient of x^n in the (2^n)-th power of 1 + Sum_{k>=0} x^(2^k), n>=0.

1, 2, 10, 112, 3636, 361088, 117079872, 129800518656, 507459311716256, 7132509596479495168, 364731747086997051317248, 68393727517839045695827152896, 47308582295958146135211324829685760

Offset

0,2

Links

Table of n, a(n) for n=0..12.

Formula

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

Example

G.f.: A(x) = 1 + 2*x + 10*x^2 + 112*x^3 + 3636*x^4 + 361088*x^5 +...
Let F(x) = 1 + x + x^2 + x^4 + x^8 + x^16 +...+ x^(2^n) +...
then A(x) = 1 + log(F(2x)) + log(F(4x))^2/2! + log(F(8x))^3/3! +...+ log(F(2^n*x))^n/n! +...
Also, coefficients in powers F(x)^(2^n) begin:
F^1: [(1),1,1,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,...];
F^2: [1,(2),3,2,3,2,2,0,3,2,2,0,2,0,0,0,3,2,2,0,2,0,0,0,...];
F^4: [1,4,(10),16,23,28,34,32,35,36,46,40,46,40,44,24,35,...];
F^8: [1,8,36,(112),274,560,1008,1632,2431,3368,4452,5664,...];
F^16: [1,16,136,800,(3636),13568,43248,121088,304066,695808,...];
F^32: [1,32,528,5952,51368,(361088),2149664,11134208,51175700,...];
F^64: [1,64,2080,45696,762448,10295040,(117079872),1152640512,...];
F^128:[1,128,8256,357632,11700384,308278784,6811832960,(129800518656), ...]; ...
where the coefficients in parenthesis form the initial terms of this sequence.

Prog

(PARI) {a(n)=local(G=1+sum(m=0, #binary(n), x^(2^m))+x*O(x^n)); polcoeff(G^(2^n), n)}
(PARI) {a(n)=local(G=1+sum(m=0, #binary(n), x^(2^m))+x*O(x^n)); polcoeff(sum(m=0, n, log(subst(G, x, 2^m*x))^m/m!), n)}

Crossrefs

Sequence in context: A234296 A049505 A136518 * A363206 A317342 A226300

Adjacent sequences: A168366 A168367 A168368 * A168370 A168371 A168372

Keyword

nonn

Author

Paul D. Hanna, Nov 24 2009

Status

approved