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A100193 a(n) = Sum_{k=0..n} binomial(2n,n+k)*3^k. 2
1, 5, 27, 146, 787, 4230, 22686, 121476, 649731, 3472382, 18546922, 99023292, 528535726, 2820451964, 15048601308, 80283276936, 428271193827, 2284478396334, 12185310873138, 64993897108236, 346655914156602 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..20.

FORMULA

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

G.f.: sqrt(1-4x)(3sqrt(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(2n,k)3^(n-k)}; - Paul Barry, Jan 11 2007

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.

Sequence in context: A293295 A015535 A026292 * A158869 A162557 A134425

Adjacent sequences:  A100190 A100191 A100192 * A100194 A100195 A100196

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Nov 08 2004

STATUS

approved

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Last modified January 18 20:57 EST 2019. Contains 319282 sequences. (Running on oeis4.)