A046748 Row sums of triangle A046521.

1, 3, 13, 61, 295, 1447, 7151, 35491, 176597, 880125, 4390901, 21920913, 109486993, 547018941, 2733608905, 13662695645, 68294088535, 341399727335, 1706739347095, 8532741458075, 42660172763995, 213287735579135, 1066389745361635, 5331765761680895

Offset: 0

Author

Wolfdieter Lang, Dec 11 1999

Keywords

• easy
• nonn

Comments

Hankel transform is A082761. - Paul Barry, Apr 14 2010

Examples

			G.f. = 1 + 3*x + 13*x^2 + 61*x^3 + 295*x^4 + 1447*x^5 + 7151*x^6 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• T.-X. He, L. W. Shapiro, Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group, Lin. Alg. Applic. 532 (2017) 25-41, p 35.

Crossrefs

Cf. A000108, A046521, A046714, A082761.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!( Sqrt(1-4*x)/(1-5*x) )); // G. C. Greubel, Jul 28 2024

Mathematica

a[ n_] := SeriesCoefficient[ Sqrt[ 1 - 4 x] / (1 - 5 x), {x, 0, n}]; (* Michael Somos, May 25 2014 *)
a[ n_] := Binomial[ 2 n, n] Hypergeometric2F1[ -n, 1, 1/2, -1/4]; (* Michael Somos, May 25 2014 *)

PARI

{a(n) = if( n<0, 0, polcoeff( sqrt( 1 - 4*x + x * O(x^n)) / (1 - 5*x), n))}; /* Michael Somos, May 25 2014 */

SageMath

def A046748_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( sqrt(1-4*x)/(1-5*x) ).list()
A046748_list(40) # G. C. Greubel, Jul 28 2024

Formula

a(n) = binomial(2*n, n)*Sum_{k=0..n} binomial(n, k)/binomial(2*k, k).

a(n) = 5^n - 2*A046714(n-1), A046714(-1) := 0.

a(n) = 5*a(n-1) - 2*A000108(n-1).

G.f.: sqrt(1-4*x)/(1-5*x).

a(n) = (3*(3*n-2)/n)*a(n-1) - (10*(2*n-3)/n)*a(n-2), n >= 1, a(-1) := 0, a(0)=1 (homogeneous recursion).

a(n) = binomial(2*n,n)*hypergeom([ -n,1 ],[ 1/2 ],-1/4) (hypergeometric 2F1 form).

0 = a(n)*(+400*a(n+1) - 330*a(n+2) + 50*a(n+3)) + a(n+1)*(-30*a(n+1) + 71*a(n+2) - 15*a(n+3)) + a(n+2)*(-3*a(n+2) + a(n+3)) for all n in Z. - Michael Somos, May 25 2014

a(n) ~ 5^(n - 1/2). - Vaclav Kotesovec, Jul 07 2016

D-finite with recurrence n*a(n) +3*(-3*n+2)*a(n-1) +10*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jul 23 2017