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A162543 A Chebyshev transform of the large Schroeder numbers A006318. 3
1, 2, 5, 18, 73, 312, 1391, 6406, 30235, 145478, 710951, 3519248, 17608681, 88914250, 452512229, 2318774506, 11953427329, 61948592936, 322570037543, 1686777086942, 8854240330363, 46638995523598, 246443050810895 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Hankel transform is the Somos-4 variant A162546.

LINKS

Fung Lam, Table of n, a(n) for n = 0..1335

FORMULA

G.f.: (1/(1+x^2))*S(x/(1+x^2)), S(x) the g.f. of A006318;

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

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

a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k,k)*A006318(n-2*k).

Recurrence: (n+1)*a(n) = (5-n)*a(n-6) + 3*(2*n-7)*a(n-5) + (11-4*n)*a(n-4)  + 12*(n-2)*a(n-3) + (5-4*n)*a(n-2) + 3*(2*n-1)*a(n-1), n>=6. - Fung Lam, Feb 19 2014

MATHEMATICA

CoefficientList[Series[(1-x+x^2 - Sqrt[1-6*x+3*x^2-6*x^3+x^4])/(2*x*(1+x^2)), {n, 0, 30}], x] (* G. C. Greubel, Feb 23 2019 *)

PROG

(PARI) my(x='x+O('x^30)); Vec((1-x+x^2 - sqrt(1-6*x+3*x^2-6*x^3+x^4))/( 2*x*(1+x^2))) \\ G. C. Greubel, Feb 23 2019

(MAGMA) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-x+x^2 - Sqrt(1-6*x+3*x^2-6*x^3+x^4))/( 2*x*(1+x^2)) )); // G. C. Greubel, Feb 23 2019

(Sage) ((1-x+x^2 -sqrt(1-6*x+3*x^2-6*x^3+x^4))/( 2*x*(1+x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 23 2019

(GAP) a:=[2, 5, 18, 73, 312, 1391];; for n in [7..30] do a[n]:=(3*(2*n-1)*a[n-1] - (4*n-5)*a[n-2] +12*(n-2)*a[n-3] -(4*n-11)*a[n-4] +3*(2*n-7)*a[n-5] -(n-5)*a[n-6])/(n+1); od; Concatenation([1], a); # G. C. Greubel, Feb 23 2019

CROSSREFS

Cf. A162548.

Sequence in context: A189843 A045612 A103940 * A039744 A319121 A289655

Adjacent sequences:  A162540 A162541 A162542 * A162544 A162545 A162546

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Jul 05 2009

STATUS

approved

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Last modified December 7 17:41 EST 2019. Contains 329847 sequences. (Running on oeis4.)