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%I A108626 #52 Oct 08 2023 09:04:22
%S A108626 1,2,5,14,41,124,383,1200,3799,12122,38919,125578,406865,1322772,
%T A108626 4313155,14099524,46192483,151628090,498578411,1641921014,5414619739,
%U A108626 17878144968,59097039545,195548471268,647665451911,2146947613286
%N A108626 Antidiagonal sums of square array A108625, in which row n equals the crystal ball sequence for A_n lattice.
%C A108626 Limit a(n+1)/a(n) = 3.3829757679... = 1/r = 3 + r + r^2, where r is radius of convergence of A(x), which diverges at x=r.
%C A108626 Infinitely many recurrence relations of even order 2d can be built for this sequence; first define the following polynomial: P(d) = (1/2^d) * Sum_{i=0..floor(d/2)} binomial(d, 2*i) * (x^4 + 2*x^2 - 4*x + 1)^i * (x^2 + 2*x - 1)^(d - 2*i)) then call c(d,k) the coefficient of term with power k in the polynomial P(d); then we have the relation: Sum_{k=0..2*d} c(d, 2*d-k)*a(n+k) = (-1)^d * Sum_{k=0..n} Sum_{i=0..k} binomial(n-k, d+i)*binomial(n-k, i)*binomial(n-i, k-i). - _Thomas Baruchel_, Jan 26 2015
%H A108626 Alois P. Heinz, <a href="/A108626/b108626.txt">Table of n, a(n) for n = 0..1000</a>
%H A108626 Thomas Baruchel and C. Elsner, <a href="http://arxiv.org/abs/1602.06445">On error sums formed by rational approximations with split denominators</a>, arXiv preprint arXiv:1602.06445 [math.NT], 2016.
%H A108626 Sergi Elizalde, <a href="https://arxiv.org/abs/2008.05669">Symmetric peaks and symmetric valleys in Dyck paths</a>, arXiv:2008.05669 [math.CO], 2020, p. 16.
%F A108626 a(n) = Sum_{k=0..n} Sum_{i=0..k} C(n, i)^2 * C(n+k-i, k-i).
%F A108626 G.f.: 1 / sqrt(x^4 + 2*x^2 - 4*x + 1). - _Thomas Baruchel_, Nov 08 2014
%F A108626 G.f.: A(x) = exp( Sum_{n>=1} A108627(n)*x^n/n ), where A108627 has g.f.: 2*x*(1 - x - x^3)/((1-x)*(1 - 3*x - x^2 - x^3)).
%F A108626 a(n) = ( (5*n-3)*a(n-1) - (6*n-8)*a(n-2) + (2*n-4)*a(n-3) - (n-2)*a(n-4) + (n-3)*a(n-5) ) / n. - _Thomas Baruchel_, Nov 08 2014
%F A108626 a(n+2) - 2*a(n+1) - a(n) = 2*Sum_{k=0..n} Sum_{i=0..k} binomial(n-k+1,i-1)*binomial(n-k+1,i)*binomial(n-i+1,k-i) = Sum_{k=0..n} a(k)*A086581(n-k+1). - _Thomas Baruchel_, Nov 08 2014
%F A108626 G.f.: Sum_{n>=0} (2*n)!/n!^2 * x^(2*n) / ((1-x)*(1-2*x)^(3*n+1)). - _Paul D. Hanna_, Nov 08 2014
%F A108626 G.f.: Sum_{n>=0} x^n/(1-x)^(n+1) * Sum_{k=0..n} C(n,k)^2 * x^k. - _Paul D. Hanna_, Nov 08 2014
%F A108626 Partial sums of A171155: a(n) = Sum_{i=0..n} A171155(n). - _Thomas Baruchel_, Nov 08 2014
%F A108626 Recurrence: n*a(n) = 2*(2*n-1)*a(n-1) - 2*(n-1)*a(n-2) - (n-2)*a(n-4). - _Vaclav Kotesovec_, Dec 20 2015
%F A108626 a(n) = Sum_{k=0..n} binomial(n,k)*hypergeometric3F2([-k,k-n,k-n], [1,-n], 1). - _Peter Luschny_, Feb 13 2018
%e A108626 Log(A(x)) = 2*x + 6*x^2/2 + 20*x^3/3 + ... + A108627(n)*x^n/n + ...
%p A108626 a := n -> add(binomial(n,k)*hypergeom([-k,k-n,k-n], [1,-n], 1), k=0..n):
%p A108626 seq(simplify(a(n)), n=0..25); # _Peter Luschny_, Feb 13 2018
%t A108626 CoefficientList[Series[1/Sqrt[x^4+2*x^2-4*x+1], {x, 0, 50}], x] (* _Vincenzo Librandi_, Nov 08 2014 *)
%o A108626 (PARI) a(n)=sum(k=0,n,sum(i=0,k,binomial(n-k,i)^2*binomial(n-i,k-i)))
%o A108626 (PARI) {a(n)=polcoeff( sum(m=0, n, x^m * sum(k=0, m, binomial(m, k)^2 * x^k) / (1-x +x*O(x^n))^(m+1)) , n)}
%o A108626 for(n=0, 30, print1(a(n), ", ")) \\ _Paul D. Hanna_, Nov 08 2014
%o A108626 (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( 1/Sqrt(1-4*x+2*x^2+x^4) )); // _G. C. Greubel_, Oct 06 2023
%o A108626 (SageMath)
%o A108626 def A108626_list(prec):
%o A108626     P.<x> = PowerSeriesRing(ZZ, prec)
%o A108626     return P( 1/sqrt(1-4*x+2*x^2+x^4) ).list()
%o A108626 A108626_list(40) # _G. C. Greubel_, Oct 06 2023
%Y A108626 Cf. A108625, A108627, A180091.
%K A108626 nonn
%O A108626 0,2
%A A108626 _Paul D. Hanna_, Jun 12 2005

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