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A046748 Row sums of triangle A046521. 6
1, 3, 13, 61, 295, 1447, 7151, 35491, 176597, 880125, 4390901, 21920913, 109486993, 547018941, 2733608905, 13662695645, 68294088535, 341399727335, 1706739347095, 8532741458075, 42660172763995, 213287735579135, 1066389745361635, 5331765761680895 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 21 03:31 EDT 2022. Contains 353887 sequences. (Running on oeis4.)