A162415 L.g.f.: Sum_{n>=1} a(n)*x^n/n = log( Sum_{n>=0} x^(2^n-1) ).

1, -1, 4, -5, 6, -10, 22, -29, 40, -66, 100, -146, 222, -344, 534, -797, 1208, -1846, 2794, -4230, 6430, -9780, 14836, -22514, 34206, -51936, 78826, -119684, 181744, -275940, 418966, -636125, 965848, -1466438, 2226482, -3380510, 5132678

Offset

1,3

Comments

Limit a(n+1)/a(n) = -1.518310626574179412829374208878425316378446155444786132846991305715550168878405863954219813056513...

Links

Robert Israel, Table of n, a(n) for n = 1..4962

Formula

a(n) = (n+1)*sum(m=1..n+1, T(n+1,m)/m*(-1)^(m+1)), where T(n,m):=(1+(-1)^(n-m))/2*sum(k=1..(n-m)/2, binomial(m,k)*T((n-m)/2,k)), T(n,n)=1. - Vladimir Kruchinin, Mar 18 2015

Example

G.f.: L(x) = x - x^2/2 + 4*x^3/3 - 5*x^4/4 + 6*x^5/5 - 10*x^6/6 +-...
where L(x) = log(1 + x + x^3 + x^7 + x^15 + x^31 +...+ x^(2^n-1) +...).

Maple

N:= 100: # to get a(1) to a(N)
L:= ln(add(x^(2^j-1), j= 0 .. ceil(log[2](N)))):
S:= series(L, x, N+1):
seq(coeff(S, x, n)*n, n=1..N); # Robert Israel, Mar 18 2015

Prog

(PARI) {a(n)=local(L=log(sum(m=0, #binary(n), x^(2^m-1))+x*O(x^n))); n*polcoeff(L, n)}
(Maxima)
T(n, m):=if n=m then 1 else (1+(-1)^(n-m))/2*sum(binomial(m, k)*T((n-m)/2, k), k, 1, (n-m)/2); makelist(n*sum(T(n, m)/m*(-1)^(m+1), m, 1, n), n, 1, 20); /* Vladimir Kruchinin, Mar 18 2015 */

Crossrefs

Cf. A162416.

Sequence in context: A101590 A057916 A249853 * A007606 A284513 A047311

Adjacent sequences: A162412 A162413 A162414 * A162416 A162417 A162418

Keyword

sign

Author

Paul D. Hanna, Jul 02 2009

Extensions

Offset corrected by Robert Israel, Mar 18 2015

Status

approved