A301584 G.f.: Sum_{n>=0} ((1+x)^(2*n) - 1)^n.

1, 2, 17, 264, 5784, 163610, 5667551, 232280480, 10991951114, 589780778314, 35379149504709, 2346218124687516, 170439977706143335, 13459938431949414118, 1148107512505151099653, 105194122765096703619248, 10303686044959088279454117, 1074408525677705370497704526, 118828297870115694372235974855, 13893778686151373846512389392672, 1712370237144948501135060958863978

Offset: 0

Author

Paul D. Hanna, Mar 24 2018

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 2*x + 17*x^2 + 264*x^3 + 5784*x^4 + 163610*x^5 + 5667551*x^6 + 232280480*x^7 + 10991951114*x^8 + 589780778314*x^9 + ...
such that
A(x) = 1 + ((1+x)^2-1) + ((1+x)^4-1)^2 + ((1+x)^6-1)^3 + ((1+x)^8-1)^4 + ((1+x)^10-1)^5 + ((1+x)^12-1)^6 + ((1+x)^14-1)^7 + ...
Also,
A(x) = 1/2 + (1+x)^2/(1 + (1+x)^2)^2 + (1+x)^8/(1 + (1+x)^4)^3 + (1+x)^18/(1 + (1+x)^6)^4 + (1+x)^32/(1 + (1+x)^8)^5 + (1+x)^50/(1 + (1+x)^10)^6 + ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..340

Crossrefs

Cf. A122400, A301585, A301586.

Programs

PARI

{a(n) = my(A,o=x*O(x^n)); A = sum(m=0,n, ((1+x +o)^(2*m) - 1)^m ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

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

a(n) ~ c * d^n * n! / sqrt(n), where d = 2*A317855 = 6.3221773077308576276603444051762649834527655483771126832545564150753941184386... and c = 0.302715376391132275494451399946850989516917... - Vaclav Kotesovec, Aug 09 2018