A188455 G.f.: 1 = Sum_{n>=0} a(n)*x^n/(1 + 2^n*x)^(2n+1).

1, 1, 5, 77, 3191, 332481, 83495679, 49089025473, 66142622730623, 200954040909283841, 1359156203343916471295, 20253823024219712679748609, 659335186924510858208484730879, 46554554840488755704034417937268737

Offset: 0

Author

Paul D. Hanna, Mar 31 2011

Keywords

• nonn

Comments

G.f. satisfies a variant of an identity of the Catalan numbers (A000108):

1 = Sum_{n>=0} A000108(n)*x^n/(1 + x)^(2n+1).

Also, g.f. satisfies a variant of an identity involving A003024:

1 = Sum_{n>=0} A003024(n)*x^n/(1 + 2^n*x)^(n+1),

where A003024(n) is the number of acyclic digraphs with n labeled nodes.

Examples

			G.f.: 1 = 1/(1+x) + x/(1+2*x)^3 + 5*x^2/(1+4*x)^5 + 77*x^3/(1+8*x)^7 + 3191*x^4/(1+16*x)^9 + 332481*x^5/(1+32*x)^11 +...
		

Crossrefs

Cf. A188456, A188457, A003024.

Programs

PARI

{a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/(1+2^k*x+x*O(x^n))^(2*k+1)), n)}