A158096 - OEIS

%I #13 Oct 03 2015 15:02:59

%S 1,2,6,188,16614,6744492,11466697660,80444371592472,

%T 2306003921102413254,268654794307394089145676,

%U 126765597337037378441876059252,241678070947171631269022075304755208,1858395916567280733577643964109494506976348,57560683587055569906379529978030563771752589955832

%N G.f.: A(x) = exp( Sum_{n>=1} x^n/n * 2^(n^2)/(1 + 2^(n^2)*x^n) ).

%C Compare to g.f. of A010054:

%C exp( Sum_{n>=1} x^n/(1 + x^n)/n ) = 1 + x + x^3 + x^6 + x^10 +...

%F G.f.: exp( Sum_{n>=1} x^n/n * Sum_{d|n} -(-1)^d * 2^(n^2/d) * d ). - _Paul D. Hanna_, Oct 02 2015

%F Logarithmic derivative equals A262826.

%e G.f.: A(x) = 1 + 2*x + 6*x^2 + 188*x^3 + 16614*x^4 + 6744492*x^5 +...

%e where

%e log(A(x)) = 2/(1 + 2*x)*x + 2^4/(1 + 2^4*x^2)*x^2/2 + 2^9/(1 + 2^9*x^3)*x^3/3 + 2^16/(1 + 2^16*x^4)*x^4/4 + 2^25/(1 + 2^25*x^5)*x^5/5 +...

%e Explicitly,

%e log(A(x)) = 2*x + 8*x^2/2 + 536*x^3/3 + 64960*x^4/4 + 33554592*x^5/5 + 68718964352*x^6/6 + 562949953422208*x^7/7 +...+ A262826(n)*x^n/n +...

%o (PARI) {a(n)=if(n==0,1,polcoeff(exp(sum(k=1,n, x^k/k * 2^(k^2)/(1 + 2^(k^2)*x^k +x*O(x^n)))),n))}

%o for(n=0,20,print1(a(n),", "))

%o (PARI) {a(n) = polcoeff( exp( sum(m=1, n, x^m/m * sumdiv(m, d, -(-1)^d * 2^(m^2/d) * d) ) +x*O(x^n)), n)}

%o for(n=0,20,print1(a(n),", ")) \\ _Paul D. Hanna_, Oct 02 2015

%Y Cf. A262826 (log), A262825, A158097, A155200.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 26 2009