A108626 Antidiagonal sums of square array A108625, in which row n equals the crystal ball sequence for A_n lattice.

1, 2, 5, 14, 41, 124, 383, 1200, 3799, 12122, 38919, 125578, 406865, 1322772, 4313155, 14099524, 46192483, 151628090, 498578411, 1641921014, 5414619739, 17878144968, 59097039545, 195548471268, 647665451911, 2146947613286

Offset: 0

Author

Paul D. Hanna, Jun 12 2005

Keywords

• nonn

Comments

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.

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

Examples

			Log(A(x)) = 2*x + 6*x^2/2 + 20*x^3/3 + ... + A108627(n)*x^n/n + ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Thomas Baruchel and C. Elsner, On error sums formed by rational approximations with split denominators, arXiv preprint arXiv:1602.06445 [math.NT], 2016.
• Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020, p. 16.

Crossrefs

Cf. A108625, A108627, A180091.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( 1/Sqrt(1-4*x+2*x^2+x^4) )); // G. C. Greubel, Oct 06 2023

Maple

a := n -> add(binomial(n,k)*hypergeom([-k,k-n,k-n], [1,-n], 1), k=0..n):
seq(simplify(a(n)), n=0..25); # Peter Luschny, Feb 13 2018

Mathematica

CoefficientList[Series[1/Sqrt[x^4+2*x^2-4*x+1], {x, 0, 50}], x] (* Vincenzo Librandi, Nov 08 2014 *)

PARI

a(n)=sum(k=0,n,sum(i=0,k,binomial(n-k,i)^2*binomial(n-i,k-i)))

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)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Nov 08 2014

SageMath

def A108626_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/sqrt(1-4*x+2*x^2+x^4) ).list()
A108626_list(40) # G. C. Greubel, Oct 06 2023

Formula

a(n) = Sum_{k=0..n} Sum_{i=0..k} C(n, i)^2 * C(n+k-i, k-i).

G.f.: 1 / sqrt(x^4 + 2*x^2 - 4*x + 1). - Thomas Baruchel, Nov 08 2014

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)).

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

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

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

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

Partial sums of A171155: a(n) = Sum_{i=0..n} A171155(n). - Thomas Baruchel, Nov 08 2014

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

a(n) = Sum_{k=0..n} binomial(n,k)*hypergeometric3F2([-k,k-n,k-n], [1,-n], 1). - Peter Luschny, Feb 13 2018