A176280 Diagonal sums of number triangle A046521.

1, 2, 7, 26, 101, 402, 1625, 6638, 27319, 113054, 469811, 1958706, 8187063, 34290934, 143864999, 604402050, 2542083509, 10702020746, 45090876913, 190110250998, 801997354525, 3384971428258, 14292950533517, 60373808435046, 255102065046401, 1078202260326002

Offset: 0

Author

Paul Barry, Apr 14 2010

Keywords

• nonn
• easy

Comments

Hankel transform is A176281.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( Sqrt(1-4*x)/(1-4*x-x^2) )); // G. C. Greubel, Nov 24 2019

Maple

seq(coeff(series(sqrt(1-4*x)/(1-4*x-x^2), x, n+1), x, n), n = 0..30);
# G. C. Greubel, Nov 24 2019
a := n -> 4^n*binomial(n-1/2, n)*hypergeom([1, (1-n)/2, -n/2], [1/2, 1/2-n], -1/4):
seq(simplify(a(n)), n = 0..25); # Peter Luschny, Mar 30 2025

Mathematica

CoefficientList[Series[Sqrt[1-4*x]/(1-4*x-x^2), {x, 0, 30}], x] (* Vaclav Kotesovec, Oct 21 2012 *)

PARI

my(x='x+O('x^30)); Vec(sqrt(1-4*x)/(1-4*x-x^2)) \\ G. C. Greubel, Nov 24 2019

Sage

def A176280_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( sqrt(1-4*x)/(1-4*x-x^2) ).list()
A176280_list(30) # G. C. Greubel, Nov 24 2019

Formula

a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*C(2*(n-k),n-k)/C(2*k,k).

From Vaclav Kotesovec, Oct 21 2012: (Start)

G.f.: sqrt(1-4*x)/(1-4*x-x^2).

Recurrence: n*a(n) = 2*(4*n-3)*a(n-1) - 3*(5*n-8)*a(n-2) - 2*(2*n-3)*a(n-3).

a(n) ~ (2+sqrt(5))^n/(2*sqrt(5)). (End)

a(n) = 4^n*binomial(n-1/2, n)*hypergeom([1, (1-n)/2, -n/2], [1/2, 1/2-n], -1/4). - Peter Luschny, Mar 30 2025