login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A084771 Coefficients of expansion of 1/sqrt(1-10*x+9*x^2); also, a(n) is the central coefficient of (1+5*x+4*x^2)^n. 10
1, 5, 33, 245, 1921, 15525, 127905, 1067925, 9004545, 76499525, 653808673, 5614995765, 48416454529, 418895174885, 3634723102113, 31616937184725, 275621102802945, 2407331941640325, 21061836725455905, 184550106298084725 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U steps come in four colors and the H steps come in five colors. - N-E. Fahssi, Mar 30 2008

Number of lattice paths from (0,0) to (n,n) using steps (1,0), (0,1), and three kinds of steps (1,1). [Joerg Arndt, Jul 01 2011]

Sums of squares of coefficients of (1+2*x)^n. [Joerg Arndt, Jul 06 2011]

The Hankel transform of this sequence gives A103488 . - Philippe Deléham, Dec 02 2007

Partial sums of A085363. - J. M. Bergot, Jun 12 2013

REFERENCES

Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.

Greene, Curtis. "Posets of shuffles." Journal of Combinatorial Theory, Series A 47.2 (1988): 191-206. See Eq. (30).

Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Christopher Huffaker, Nathan McCue, Cameron N. Miller, Kayla S. Miller, The M&M Game: From Morsels to Modern Mathematics, arXiv:1508.06542 [math.HO], (24-August-2015)

Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

FORMULA

G.f.: 1/sqrt(1-10*x+9*x^2).

Binomial transform of A059304. G.f.: Sum_{k>=0} binomial(2*k, k)*(2*x)^k/(1-x)^(k+1). E.g.f.: exp(5*x)*BesselI(0, 4*x). - Vladeta Jovovic, Aug 20 2003

a(n) = sum(k=0..n, sum(j=0..n-k, C(n,j)*C(n-j,k)*C(2*n-2*j,n-j) ) ). - Paul Barry, May 19 2006

a(n) = sum(k=0..n, 4^k*(C(n,k))^2 ) [From heruneedollar (heruneedollar(AT)gmail.com), Mar 20 2010]

Asymptotic: a(n) ~ 3^(2*n+1)/(2*sqrt(2*Pi*n)). [Vaclav Kotesovec, Sep 11 2012]

Conjecture: n*a(n) +5*(-2*n+1)*a(n-1) +9*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 26 2012

a(n) = hypergeom([-n,1/2], [1], -8). - Peter Luschny, Apr 26 2016

EXAMPLE

G.f.: 1/sqrt(1-2*b*x+(b^2-4*c)*x^2) yields central coefficients of (1+b*x+c*x^2)^n.

MAPLE

seq(simplify(hypergeom([-n, 1/2], [1], -8)), n=0..19); # Peter Luschny, Apr 26 2016

MATHEMATICA

Table[n! SeriesCoefficient[E^(5 x) BesselI[0, 4 x], {x, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, May 10 2013 *)

Table[Hypergeometric2F1[-n, -n, 1, 4], {n, 0, 19}] (* Vladimir Reshetnikov, Nov 29 2013 *)

CoefficientList[Series[1/Sqrt[1-10x+9x^2], {x, 0, 30}], x] (* Harvey P. Dale, Mar 08 2016 *)

PROG

(PARI) a(n)=sum(k=0, n, binomial(n, k)^2*4^k)

(PARI) a(n)=if(n<0, 0, polcoeff((1+5*x+4*x^2)^n, n))

(PARI) /* as lattice paths: same as in A092566 but use */

steps=[[1, 0], [0, 1], [1, 1], [1, 1], [1, 1]]; /* note the triple [1, 1] */

/* Joerg Arndt, Jul 01 2011 */

(PARI) a(n)={local(v=Vec((1+2*x)^n)); sum(k=1, #v, v[k]^2); } /* Joerg Arndt, Jul 06 2011 */

(PARI) a(n)={local(v=Vec((1+2*I*x)^n)); sum(k=1, #v, real(v[k])^2+imag(v[k])^2); } /* Joerg Arndt, Jul 06 2011 */

CROSSREFS

Cf. A246923 (a(n)^2).

Sequence in context: A093427 A142989 A084131 * A153398 A242522 A034015

Adjacent sequences:  A084768 A084769 A084770 * A084772 A084773 A084774

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 10 2003

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 29 03:29 EDT 2017. Contains 284250 sequences.