A156101 L.g.f.: A(x) = Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} (1 + 2^n*x)^n*x^n/n .

1, 5, 13, 65, 401, 3521, 43457, 738305, 17746177, 593695745, 27878501377, 1840450134017, 169904883945473, 22139372291866625, 4036405254299041793, 1038968242677362458625, 375102612647535161966593

Offset: 1

Author

Paul D. Hanna, Feb 04 2009

Keywords

• nonn

Comments

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

Examples

			G.f.: A(x) = x + 5*x^2/2 + 13*x^3/3 + 65*x^4/4 + 401*x^5/5 + ...
A(x) = (1 + 2*x)*x + (1 + 2^2*x)^2*x^2/2 + (1 + 2^3*x)^3*x^3/3 + ...
exp(A(x)) = 1 + x + 3*x^2 + 7*x^3 + 25*x^4 + 113*x^5 + 741*x^6 + ...
		

Crossrefs

Cf. A156100.

Programs

Mathematica

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 *)

PARI

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

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

Formula

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

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

Extensions

Offset corrected by Vaclav Kotesovec, Mar 06 2014