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A081671 Expansion of e.g.f. exp(4x) * I_0(2x). 28
1, 4, 18, 88, 454, 2424, 13236, 73392, 411462, 2325976, 13233628, 75682512, 434662684, 2505229744, 14482673832, 83940771168, 487610895942, 2838118247064, 16547996212044, 96635257790352, 565107853947444, 3308820294176016, 19395905063796312, 113814537122646432 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Binomial transform of A026375. Second binomial transform of A000984.

Largest coefficient of (1+4x+x^2)^n. - Paul Barry, Dec 15 2003

Row sums of triangle in A124574 . - Philippe Deléham, Sep 27 2007

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps come in 4 colors. - N-E. Fahssi, Feb 05 2008

Diagonal of rational function 1/(1 - (x^2 + 4*x*y + y^2)). - Gheorghe Coserea, Aug 01 2018

LINKS

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

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arXiv preprint arXiv:1110.6638 [math.NT], 2011.

Francesc Fite and Andrew V. Sutherland, Sato-Tate distributions of twists of y^2= x^5-x and y^2= x^6+1, arXiv preprint arXiv:1203.1476 [math.NT], 2012. - From N. J. A. Sloane, Sep 14 2012

Rigoberto Flórez, Leandro Junes, José L. Ramírez, Further Results on Paths in an n-Dimensional Cubic Lattice, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.2.

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

FORMULA

a(n) = sum{m=0..n, sum{k=0..m, C(n, m)C(m, k)C(2k, k)}}.

G.f.: 1/sqrt((1-2*x)*(1-6*x)). - Vladeta Jovovic, Oct 09 2003

a(n) = Sum_{k=0..n} 2^(n-k) * C(n, k) * C(2*k, k). - Paul Barry, Jan 27 2005

a(n) = Sum_{k=0..n} 6^(n-k) * (-1)^k * C(n,k) * C(2*k,k). - Paul D. Hanna, Dec 09 2018

Recurrence: n*a(n) = 4*(2*n-1)*a(n-1) - 12*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 13 2012

a(n) ~ sqrt(3/(2*Pi*n))*6^n. - Vaclav Kotesovec, Oct 13 2012

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

a(n) = GegenbauerC(n, -n, -2). - Peter Luschny, May 09 2016

a(n) = Sum_{k=0..floor(n/2)} 4^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). - Seiichi Manyama, May 04 2019

MAPLE

seq(simplify(2^n*hypergeom([-n, 1/2], [1], -2)), n=0..23); # Peter Luschny, Apr 26 2016

seq(simplify(GegenbauerC(n, -n, -2)), n=0..23); # Peter Luschny, May 09 2016

MATHEMATICA

Table[SeriesCoefficient[1/Sqrt[(1-2*x)*(1-6*x)], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 13 2012 *)

PROG

(Maxima) a(n):=coeff(expand((1+4*x+x^2)^n), x^n);

makelist(a(n), n, 0, 30); /* Emanuele Munarini, Apr 27 2012 */

(PARI) x='x+O('x^66); Vec(1/sqrt((1-2*x)*(1-6*x))) \\ Joerg Arndt, May 07 2013

(PARI) {a(n) = sum(k=0, n\2, 4^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k))} \\ Seiichi Manyama, May 04 2019

CROSSREFS

Column 4 of A292627.

Sequence in context: A083325 A050146 A083879 * A244785 A260650 A006629

Adjacent sequences:  A081668 A081669 A081670 * A081672 A081673 A081674

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Mar 28 2003

STATUS

approved

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Last modified June 20 09:27 EDT 2019. Contains 324234 sequences. (Running on oeis4.)