A100193 a(n) = Sum_{k=0..n} binomial(2n,n+k)*3^k.

1, 5, 27, 146, 787, 4230, 22686, 121476, 649731, 3472382, 18546922, 99023292, 528535726, 2820451964, 15048601308, 80283276936, 428271193827, 2284478396334, 12185310873138, 64993897108236, 346655914156602

Offset

0,2

Comments

A transform of 3^n under the mapping g(x)->(1/sqrt(1-4x))g(x*c(x)^2), where c(x) is the g.f. of the Catalan numbers A000108. A transform of 4^n under the mapping g(x)->(1/(c(x)*sqrt(1-4x))g(x*c(x)).

Hankel transform is A127357. In general, the Hankel transform of Sum_{k=0..n} C(2n,k)*r^(n-k) is the sequence with g.f. 1/(1-2x+r^2*x^2). - Paul Barry, Jan 11 2007

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

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

G.f.: sqrt(1-4x)*(3*sqrt(1-4x)-8x+3)/((1-4x)(6-32x)).

a(n) = Sum_{k=0..n} binomial(2n, n-k)*3^k.

a(n) = (Sum_{k=0..n} binomial(2n, n-k))*(Sum_{j=0..n} binomial(n, j)*(-1)^(n-j)*4^j).

a(n) = Sum_{k=0..n} C(n+k-1,k)*4^(n-k). - Paul Barry, Sep 28 2007

Conjecture: 9*n*a(n) + 6*(11-18*n)*a(n-1) + 16*(26*n-37)*a(n-2) + 256*(5-2*n)*a(n-3) = 0. - R. J. Mathar, Nov 09 2012

a(n) ~ (16/3)^n. - Vaclav Kotesovec, Feb 03 2014

a(n) = [x^n] 1/((1 - x)^n*(1 - 4*x)). - Ilya Gutkovskiy, Oct 12 2017

Mathematica

Table[Binomial[2*n, n]*Hypergeometric2F1[1, -n, 1+n, -3], {n, 0, 20}] (* Vaclav Kotesovec, Feb 03 2014 *)

Crossrefs

Cf. A032443, A100192, A000108, A127357.

Sequence in context: A343208 A015535 A026292 * A158869 A162557 A134425

Adjacent sequences: A100190 A100191 A100192 * A100194 A100195 A100196

Keyword

easy,nonn

Author

Paul Barry, Nov 08 2004

Status

approved