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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
OFFSET
0,2
COMMENTS
Image of A006318 under the Riordan array (1/(1-x), x/(1-x)^2).
LINKS
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
The above recurrence follows from the differential equation (4*x^4 - 14*x^3 + 15*x^2 - 7*x + 1)*A(x) - (x^6 - 11*x^5 + 29*x^4 - 29*x^3 + 11*x^2 - x)*A'(x) + x^4 - x^3 + x - 1 = 0 satisfied by the g.f. A(x). - Peter Bala, Sep 15 2024
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
Sequence in context: A141342 A232222 A241598 * A009102 A080227 A199143
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Jan 29 2009
STATUS
approved