A156101 - OEIS

%I #10 Feb 04 2018 03:20:14

%S 1,5,13,65,401,3521,43457,738305,17746177,593695745,27878501377,

%T 1840450134017,169904883945473,22139372291866625,4036405254299041793,

%U 1038968242677362458625,375102612647535161966593

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

%C Compare to l.g.f. Sum_{m>=1} (1 + x)^m * x^m/m of the Fibonacci sequence.

%F L.g.f.: A(x) = log(G(x)) where G(x) is the g.f. of A156100.

%F a(n) = n*Sum_{k=0..floor(n/2)} C(n-k,k)*2^(k(n-k))/(n-k). - _Paul D. Hanna_, Apr 10 2009

%e G.f.: A(x) = x + 5*x^2/2 + 13*x^3/3 + 65*x^4/4 + 401*x^5/5 + ...

%e A(x) = (1 + 2*x)*x + (1 + 2^2*x)^2*x^2/2 + (1 + 2^3*x)^3*x^3/3 + ...

%e exp(A(x)) = 1 + x + 3*x^2 + 7*x^3 + 25*x^4 + 113*x^5 + 741*x^6 + ...

%t Table[n*Sum[Binomial[n-k,k]*2^(k(n-k))/(n-k),{k,0,Floor[n/2]}],{n,1,20}] (* _Vaclav Kotesovec_, Mar 06 2014 *)

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

%o (PARI) {a(n)=n*sum(k=0,n\2,binomial(n-k,k)*2^(k*(n-k))/(n-k))} \\ _Paul D. Hanna_, Apr 10 2009

%Y Cf. A156100.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Feb 04 2009

%E Offset corrected by _Vaclav Kotesovec_, Mar 06 2014