A166228 Alternating sum of large Schroeder numbers.

1, 1, 5, 17, 73, 321, 1485, 7073, 34513, 171585, 866133, 4427313, 22870425, 119208321, 626178717, 3311424321, 17615732385, 94202293633, 506116560293, 2730607756881, 14788011564009, 80361643637953, 438070231780973

Offset: 0

Author

Paul Barry, Oct 09 2009

Keywords

• easy
• nonn

Comments

Hankel transform is A166231. Binomial transform is A166229.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300

Programs

Mathematica

CoefficientList[Series[(1-x-Sqrt[1-6*x+x^2])/(2*x*(1+x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

Formula

G.f.: (1-x-sqrt(1-6x+x^2))/(2x(1+x));

a(n) = Sum{k=0..n} (-1)^k*A006318(n-k) = Sum_{k=0..n} (-1)^(n-k)*A006318(k).

Conjecture: (n+1)*a(n) +(4-5n)*a(n-1) +(1-5n)*a(n-2) +(n-2)*a(n-3)=0. - R. J. Mathar, Nov 17 2011

a(n) ~ sqrt(48+34*sqrt(2))*(3+2*sqrt(2))^n/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 20 2012