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A sequence with Somos-4 Hankel transform.
3

%I #34 Nov 04 2024 16:23:56

%S 1,1,2,3,7,13,31,65,156,351,849,1993,4866,11733,28921,70997,176560,

%T 438979,1100302,2761797,6969909,17625015,44742636,113822415,290416803,

%U 742486655,1902767481,4885201701,12567065582,32382099109,83580301371

%N A sequence with Somos-4 Hankel transform.

%C Hankel transform is the Somos-4 sequence A006720(n+2).

%C The generating function A(x) satisfies A(x) = 1 + x + x^2*A(x) + (x*A(x))^2.

%C BINOMIAL transform is A087626. HANKEL transform with a(0) omitted is A051138(n+2). - _Michael Somos_, Jan 11 2013

%H G. C. Greubel, <a href="/A171416/b171416.txt">Table of n, a(n) for n = 0..1000</a>

%H Paul Barry, <a href="https://arxiv.org/abs/1912.01126">Riordan arrays, the A-matrix, and Somos 4 sequences</a>, arXiv:1912.01126 [math.CO], 2019.

%F G.f.: (1 - x^2 - sqrt(1-6*x^2-4*x^3+x^4))/(2*x^2).

%F G.f.: (1/(1-x))*c(x^2/((1-x)*(1-x^2))) where c(x) is the g.f. of A000108.

%F G.f.: 1/(1-x-x^2/(1-x^2-x^2/(1-x-x^2/(1-x^2-x^2/(1-x-x^2/(1-x^2-x^2/(1-...))))))) (continued fraction).

%F a(n) = a(n-2) + Sum_{k=1..n-1} a(k-1)*a(n-k-1) with a(0)=a(1)=1.

%F Conjecture: (n+2)*a(n) +(n+1)*a(n-1) +6*(1-n)*a(n-2) +2*(11-5*n)*a(n-3) +(10-3*n)*a(n-4) +(n-5)*a(n-5)=0. - _R. J. Mathar_, Jul 24 2012

%F G.f.: 2*(1 + x) / (1 - x^2 + sqrt(1 - 6*x^2 - 4*x^3 + x^4)).

%F (n+2) * a(n) - (6*n-6) * a(n-2) - (4*n-10) * a(n-3) + (n-4) * a(n-4) = 0 if n>3. - _Michael Somos_, Jan 11 2013

%F If we write the generating function as 1/(1-b_{0}*x/(1-c_{0}x/(1-b_{1}*x/(1-c_{1}*x/(1-...))))), then b_{n}*c_{n} = A006720(n+1)*A006720(n+3)/A006720(n+2)^2 = A377264(n)/A006720(n+2)^2. - _Thomas Scheuerle_, Oct 22 2024

%e G.f. = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 13*x^5 + 31*x^6 + 65*x^7 + 156*x^8 + ...

%p m:=30; S:=series((1 -x^2 -sqrt(1-6*x^2-4*x^3+x^4))/(2*x^2), x, m+1): seq(coeff(S, x, j), j=0..m); # _G. C. Greubel_, Feb 18 2020

%t CoefficientList[Series[(1-x^2 -Sqrt[1 -6x^2 -4x^3 +x^4])/(2x^2), {x, 0, 30}], x] (* Or *)

%t a[n_]:= a[n]= a[n-2] + Sum[a[k-1]a[n-k-1], {k, n-1}]; a[0]=a[1]=1; Array[a, 31, 0] (* _Robert G. Wilson v_, Mar 28 2011 *)

%o (PARI) {a(n) = if( n<0, 0, polcoeff( 2*(1 + x) / (1 - x^2 + sqrt(1 - 6*x^2 - 4*x^3 + x^4 + x*O(x^n))), n))}; /* _Michael Somos_, Jan 11 2013 */

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!((1-x^2-Sqrt(1-6*x^2-4*x^3+x^4))/(2*x^2))); // _G. C. Greubel_, Sep 22 2018

%o (Sage)

%o def A171416_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( (1 -x^2 -sqrt(1-6*x^2-4*x^3+x^4))/(2*x^2) ).list()

%o A171416_list(30) # _G. C. Greubel_, Feb 18 2020

%Y Cf. A006720, A051138, A087626, A377264.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Dec 08 2009