A103973 Expansion of (sqrt(1-8*x^2)+8*x^2+2*x-1)/(2*x*sqrt(1-8*x^2)).

1, 2, 4, 4, 24, 16, 160, 80, 1120, 448, 8064, 2688, 59136, 16896, 439296, 109824, 3294720, 732160, 24893440, 4978688, 189190144, 34398208, 1444724736, 240787456, 11076222976, 1704034304, 85201715200, 12171673600, 657270374400

Offset: 0

Author

Paul Barry, Feb 23 2005

Keywords

• easy
• nonn

Crossrefs

Cf. A025225, A059304.

Formula

G.f.: 1/sqrt(1-8*x^2)+(1-sqrt(1-8*x^2))/(2*x).

a(n) = sum{k=0..floor(n/2), 2^(n-k) * A000108(k) * C(k+1, n-k)}.

Conjecture D-finite with recurrence: 11*n*(n+1)*a(n)+4*n*(4*n+1)*a(n-1) +8*(27-11*n^2)*a(n-2) -32*(4*n+9)*(n-3)*a(n-3)=0. - R. J. Mathar, Nov 09 2012

a(n) ~ 2^((3*n + 1)/2) / sqrt(Pi*n) if n is even and a(n) ~ 2^((3*n + 2)/2) / (sqrt(Pi)*n^(3/2)) if n is odd. - Vaclav Kotesovec, Nov 19 2021