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

0,2

Comments

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

Links

Table of n, a(n) for n=0..23.

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.

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

Example

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

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 *)

Prog

(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 */

Crossrefs

Cf. A000108, A046521, A046714.

Sequence in context: A101368 A341250 A026704 * A256333 A200215 A074548

Adjacent sequences: A046745 A046746 A046747 * A046749 A046750 A046751

Keyword

easy,nonn

Author

Wolfdieter Lang, Dec 11 1999

Status

approved