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

0,2

Comments

Hankel transform is A174809.

Links

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

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

Example

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

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 *)

Prog

(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
(Magma) m:=30; R<x>:=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

Crossrefs

Cf. A174809.

Sequence in context: A151305 A150901 A046649 * A109972 A019023 A343523

Adjacent sequences: A174805 A174806 A174807 * A174809 A174810 A174811

Keyword

easy,nonn

Author

Paul Barry, Mar 29 2010

Status

approved