A174808 A transform of the large Schroeder numbers A006318.

1, 2, 8, 34, 162, 820, 4338, 23694, 132612, 756594, 4384022, 25729336, 152627730, 913674362, 5512542128, 33486653154, 204639278346, 1257199799116, 7760098104882, 48102326710998, 299309479778956, 1868853597670754

Offset: 0

Author

Paul Barry, Mar 29 2010

Keywords

• easy
• nonn

Comments

Hankel transform is A174809.

Examples

			G.f. = 1 + 2*x + 8*x^2 + 34*x^3 + 162*x^4 + 820*x^5 + 4338*x^6 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A174809.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-x-x^2-Sqrt(1-6*x-5*x^2+2*x^3+x^4))/(2*x*(1+x)))); // G. C. Greubel, Sep 22 2018

Maple

A174808 := proc(n)
    add(binomial(k,n-k)*A006318(k),k=0..n) ;
end proc: # R. J. Mathar, Feb 10 2015

Mathematica

CoefficientList[Series[(1-x-x^2 -Sqrt[1-6*x-5*x^2+2*x^3+x^4])/(2*x*(1 + x)), {x, 0, 30}], x] (* G. C. Greubel, Sep 22 2018 *)

PARI

x='x+O('x^30); Vec((1-x-x^2-sqrt(1-6*x-5*x^2+2*x^3+x^4))/(2*x*(1+x))) \\ G. C. Greubel, Sep 22 2018

Formula

G.f.: (1-x-x^2-sqrt(1-6*x-5*x^2+2*x^3+x^4))/(2*x*(1+x)).

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

a(n) = Sum_{k=0..n} C(k,n-k)*A006318(k).

G.f.: 1 / (1 - (x + x^2)*(1 + 1 / (1 - (x + x^2)*(1 + 1 / ...)))). - Michael Somos, Mar 30 2014

Conjecture: (n+1)*a(n) +(-5*n+4)*a(n-1) +(-11*n+13)*a(n-2) +3*(-n+1)*a(n-3) +3*(n-4)*a(n-4) +(n-5)*a(n-5)=0. - R. J. Mathar, Feb 10 2015