A166694 A transform of the large Schroeder numbers, A006318.

1, 2, 10, 52, 290, 1706, 10440, 65822, 424710, 2791340, 18622510, 125791894, 858621680, 5913143706, 41036613570, 286702877908, 2014876764170, 14234073943986, 101025202379480, 720017430722598, 5151008515543790

Offset: 0

Author

Paul Barry, Oct 18 2009

Keywords

• easy
• nonn

Comments

Apply the Riordan array (1,x/(1-x)^2) to the large Schroeder numbers.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500

Programs

Mathematica

CoefficientList[Series[(1 - 3*t + t^2 - Sqrt[1 - 10*t + 19*t^2 - 10*t^3 + t^4])/(2*t), {t, 0, 50}], t] (* G. C. Greubel, May 23 2016 *)

Formula

G.f.: (1-3x+x^2-sqrt(1-10x+19x^2-10x^3+x^4))/(2x);

G.f.: 1/(1-2x/((1-x)^2-x/(1-2x/((1-x)^2-x/(1-2x/((1-x)^2-x/(1-2x/(1-... (continued fraction);

a(n) = sum{k=0..n, (0^(n+k)+C(n+k-1,2k-1))*A006318(k)}=0^n + sum{k=0..n, C(n+k-1,2k-1)*A006318(k)}.

Conjecture: (n+1)*a(n) +5*(1-2n)*a(n-1) +19*(n-2)*a(n-2) +5*(7-2*n)*a(n-3) +(n-5)*a(n-4)=0. - R. J. Mathar, Nov 16 2011