cp's OEIS Frontend

A155867 A 'Morgan Voyce' transform of the large Schroeder numbers A006318.

%I A155867 #25 Dec 23 2024 11:35:26
%S A155867 1,3,13,65,355,2061,12501,78323,503033,3294373,21916883,147708777,
%T A155867 1006330457,6919474163,47956087733,334658965641,2349535729811,
%U A155867 16583609673797,117608812053277,837626242775875,5988634758319665
%N A155867 A 'Morgan Voyce' transform of the large Schroeder numbers A006318.
%C A155867 Image of A006318 under the Riordan array (1/(1-x), x/(1-x)^2).
%H A155867 G. C. Greubel, <a href="/A155867/b155867.txt">Table of n, a(n) for n = 0..1000</a>
%H A155867 Veronica Bitonti, Bishal Deb, and Alan D. Sokal, <a href="https://arxiv.org/abs/2412.10214">Thron-type continued fractions (T-fractions) for some classes of increasing trees</a>, arXiv:2412.10214 [math.CO], 2024. See p. 58.
%F A155867 G.f.: (1 - 3*x + x^2 - sqrt(1 - 10*x + 19*x^2 - 10*x^3 + x^4))/(2*x*(1-x)).
%F A155867 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).
%F A155867 a(n) = Sum_{k=0..n} binomial(n+k,2k)*A006318(k).
%F A155867 a(n) = Sum_{k=0..n} A085478(n,k)*A006318(k). - _Philippe Deléham_, Jan 31 2009
%F A155867 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
%F A155867 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
%t A155867 A006318[n_]:= 2*Hypergeometric2F1[-n+1, n+2, 2, -1];
%t A155867 A155867[n_]:= Sum[Binomial[n+j, 2*j]*A006318[j], {j,0,n}];
%t A155867 Table[A155867[n], {n, 0, 40}] (* _G. C. Greubel_, Jun 09 2021 *)
%o A155867 (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
%o A155867 (Sage)
%o A155867 def A155867_list(prec):
%o A155867     P.<x> = PowerSeriesRing(ZZ, prec)
%o A155867     return P( (1-3*x+x^2 -sqrt(1-10*x+19*x^2-10*x^3+x^4))/(2*x*(1-x)) ).list()
%o A155867 A155867_list(40) # _G. C. Greubel_, Jun 09 2021
%Y A155867 Cf. A006318, A085478.
%K A155867 easy,nonn
%O A155867 0,2
%A A155867 _Paul Barry_, Jan 29 2009