Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #14 Sep 08 2022 08:45:41
%S 1,0,1,2,6,17,51,156,488,1552,5006,16337,53849,179015,599535,2020924,
%T 6851150,23344138,79902364,274606264,947240592,3278404274,11381240074,
%U 39621423949,138288477617,483805404673,1696318159457,5959737806635
%N Transform of Catalan numbers whose Hankel transform gives the Somos-4 sequence.
%C Image of the Catalan numbers A000108 by the Riordan array (1-x,x(1-x^2)). Hankel transform is A006720(n+1).
%C 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 Hankel transform of a(n+1) is -A051138(n). - _Michael Somos_, Feb 10 2015
%H G. C. Greubel, <a href="/A157002/b157002.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 G.f.: (1 - sqrt(1-4*x*(1-x^2)))/(2*x*(1+x)).
%F a(n) = Sum_{k=0..n} (-1)^floor((n-k+1)/2)*C(k,floor((n-k)/2))*A000108(k).
%F 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 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 G.f. = 1 + x^2 + 2*x^3 + 6*x^4 + 17*x^5 + 51*x^6 + 156*x^7 + 488*x^8 + ...
%t CoefficientList[Series[(1-Sqrt[1-4x(1-x^2)])/(2x(1+x)), {x,0,30}], x] (* _G. C. Greubel_, Feb 26 2019 *)
%o (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 (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 (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 Cf. A000108, A006720, A025262, A051138, A056010.
%K easy,nonn
%O 0,4
%A _Paul Barry_, Feb 20 2009