A144691 Limit of the coefficient of x^(2^m+n) in B(x)^(n+1)/(n+1) as m grows, where B(x) = Sum_{k>=0} x^(2^k).

1, 1, 2, 4, 26, 106, 816, 4292, 90162, 715138, 10275886, 87498566, 1944309280, 20988667064, 380829128200, 4301687654136, 219999839271970, 3375111608092354, 90438559754079802, 1341646116200287978, 52342848299405537114, 921821277222438350170

Offset: 0

Author

Paul D. Hanna, Oct 10 2008

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 26*x^4 + 106*x^5 + 816*x^6 +...
A(x/G(x)) = G(x) = x/Series_Reversion[x*A(x)], where
G(x) = 1 + x + x^2 + 17*x^4 + 408*x^6 + 69473*x^8 + 6018928*x^10 +...
and G(x) appears to continue with only even powers of x (cf. A144692).
The inverse binomial transform forms the g.f. of A202582:
A(x/(1+x))/(1+x) = 1 + x^2 + 19*x^4 + 515*x^6 + 74383*x^8 + 6816465*x^10 +...+ A202582(n)*x^n +...
		

Links

• Max Alekseyev, Table of n, a(n) for n = 0..27

Crossrefs

Cf. A007178, A144690, A144692, A202582.

Programs

PARI

{ a(n) = local(m=n+log(n+.5)\log(2), B=sum(k=0,m,x^(2^k))); if(n<0, 0, polcoeff((B+O(x^(2^m+n+1)))^(n+1)/(n+1),2^m+n)) }

Formula

a(n) = A144690(n)/(n+1).

G.f. A(x) satisfies: A(x/(1+x))/(1+x) is an even function; i.e., the inverse binomial transform yields A202582.

Extensions

a(14), a(15) corrected and a(16)-a(23) added by Max Alekseyev, May 03 2011

a(24)-a(27) in b-file from Max Alekseyev, Dec 19 2011