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A155867 A 'Morgan Voyce' transform of the large Schroeder numbers A006318. 1
1, 3, 13, 65, 355, 2061, 12501, 78323, 503033, 3294373, 21916883, 147708777, 1006330457, 6919474163, 47956087733, 334658965641, 2349535729811, 16583609673797, 117608812053277, 837626242775875, 5988634758319665 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Image of A006318 under the Riordan array (1/(1-x), x/(1-x)^2).

LINKS

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

FORMULA

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

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

a(n) = Sum_{k=0..n} binomial(n+k,2k)*A006318(k).

a(n) = Sum_{k=0..n} A085478(n,k)*A006318(k). - Philippe Deléham, Jan 31 2009

Conjecture: (n+1)*a(n) + (4-11*n)*a(n-1) + (29*n-43)*a(n-2) +(73-29*n)*a(n-3) + (11*n-40)*a(n-4) + (5-n)*a(n-5) = 0. - R. J. Mathar, Jul 24 2012

MATHEMATICA

A006318[n_]:= 2*Hypergeometric2F1[-n+1, n+2, 2, -1];

A155867[n_]:= Sum[Binomial[n+j, 2*j]*A006318[j], {j, 0, n}];

Table[A155867[n], {n, 0, 40}] (* G. C. Greubel, Jun 09 2021 *)

PROG

(MAGMA) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-3*x+x^2 -Sqrt(1-10*x+19*x^2-10*x^3+x^4))/(2*x*(1-x)) )); // G. C. Greubel, Jun 09 2021

(Sage)

def A155867_list(prec):

    P.<x> = PowerSeriesRing(ZZ, prec)

    return P( (1-3*x+x^2 -sqrt(1-10*x+19*x^2-10*x^3+x^4))/(2*x*(1-x)) ).list()

A155867_list(40) # G. C. Greubel, Jun 09 2021

CROSSREFS

Cf. A006318, A085478.

Sequence in context: A141342 A232222 A241598 * A009102 A080227 A199143

Adjacent sequences:  A155864 A155865 A155866 * A155868 A155869 A155870

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Jan 29 2009

STATUS

approved

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Last modified August 6 00:46 EDT 2021. Contains 346493 sequences. (Running on oeis4.)