cp's OEIS Frontend

A157002 Transform of Catalan numbers whose Hankel transform gives the Somos-4 sequence.

%I A157002 #14 Sep 08 2022 08:45:41
%S A157002 1,0,1,2,6,17,51,156,488,1552,5006,16337,53849,179015,599535,2020924,
%T A157002 6851150,23344138,79902364,274606264,947240592,3278404274,11381240074,
%U A157002 39621423949,138288477617,483805404673,1696318159457,5959737806635
%N A157002 Transform of Catalan numbers whose Hankel transform gives the Somos-4 sequence.
%C A157002 Image of the Catalan numbers A000108 by the Riordan array (1-x,x(1-x^2)). Hankel transform is A006720(n+1).
%C A157002 The sequence a(n)+a(n+1) begins 1,1,3,8,23,68,... which is A056010. The sequence a(n)+a(n-1) begins 1,1,1,3,8,23,68,... which is A025262. This is obtained by applying (1-x^2,x(1-x^2)) to the Catalan numbers.
%C A157002 Hankel transform of a(n+1) is -A051138(n). - _Michael Somos_, Feb 10 2015
%H A157002 G. C. Greubel, <a href="/A157002/b157002.txt">Table of n, a(n) for n = 0..1000</a>
%H A157002 Gouce Xin, <a href="https://doi.org/10.1016/j.aam.2008.04.003">Proof of the Somos-4 Hankel determinants conjecture</a>, Advances in Applied Mathematics, Volume 42, Issue 2, February 2009, Pages 152-156.
%F A157002 G.f.: (1 - sqrt(1-4*x*(1-x^2)))/(2*x*(1+x)).
%F A157002 a(n) = Sum_{k=0..n} (-1)^floor((n-k+1)/2)*C(k,floor((n-k)/2))*A000108(k).
%F A157002 Conjecture: (n+1)*a(n) +3*(-n+1)*a(n-1) +2*(-2*n+1)*a(n-2) +2*(2*n-7)*a(n-3) +2*(2*n-7)*a(n-4)=0. - _R. J. Mathar_, Nov 19 2014
%F A157002 0 = a(n)*(+16*a(n+1) + 16*a(n+2) - 64*a(n+3) - 42*a(n+4) + 22*a(n+5)) + a(n+1)*(+16*a(n+1) + 48*a(n+2) - 46*a(n+3) - 56*a(n+4) + 22*a(n+5)) + a(n+2)*(+32*a(n+2) + 34*a(n+3) - 8*a(n+4) - 10*a(n+5)) + a(n+3)*(+18*a(n+3) + 11*a(n+4) - 9*a(n+5)) + a(n+4)*(+3*a(n+4) + a(n+5)) for all n in Z. - _Michael Somos_, Feb 10 2015
%e A157002 G.f. = 1 + x^2 + 2*x^3 + 6*x^4 + 17*x^5 + 51*x^6 + 156*x^7 + 488*x^8 + ...
%t A157002 CoefficientList[Series[(1-Sqrt[1-4x(1-x^2)])/(2x(1+x)), {x,0,30}], x] (* _G. C. Greubel_, Feb 26 2019 *)
%o A157002 (PARI) {a(n) = if( n<0, -(-1)^n / 2 * (n<-1), polcoeff( (1 - sqrt(1 - 4*x * (1 - x^2) + x^2 * O(x^n))) / (2 * x * (1 + x)), n))}; /* _Michael Somos_, Feb 10 2015 */
%o A157002 (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1 -Sqrt(1-4*x*(1-x^2)))/(2*x*(1+x)) )); // _G. C. Greubel_, Feb 26 2019
%o A157002 (Sage) ((1-sqrt(1-4*x*(1-x^2)))/(2*x*(1+x))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Feb 26 2019
%Y A157002 Cf. A000108, A006720, A025262, A051138, A056010.
%K A157002 easy,nonn
%O A157002 0,4
%A A157002 _Paul Barry_, Feb 20 2009