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
Keywords
Examples
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) +...).
Links
- Robert Israel, Table of n, a(n) for n = 1..4962
Crossrefs
Cf. A162416.
Programs
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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
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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 */
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PARI
{a(n)=local(L=log(sum(m=0,#binary(n),x^(2^m-1))+x*O(x^n)));n*polcoeff(L,n)}
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
Extensions
Offset corrected by Robert Israel, Mar 18 2015
Comments