A081671 Expansion of e.g.f. exp(4x) * I_0(2x).

1, 4, 18, 88, 454, 2424, 13236, 73392, 411462, 2325976, 13233628, 75682512, 434662684, 2505229744, 14482673832, 83940771168, 487610895942, 2838118247064, 16547996212044, 96635257790352, 565107853947444, 3308820294176016, 19395905063796312, 113814537122646432

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

Hacène Belbachir, Abdelghani Mehdaoui and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.

Isaac DeJager, Madeleine Naquin and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.

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 and 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

D-finite with 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

a(n) = (1/Pi) * Integral_{x = -1..1} (2 + 4*x^2)^n/sqrt(1 - x^2) dx = (1/Pi) * Integral_{x = -1..1} (6 - 4*x^2)^n/sqrt(1 - x^2) dx . - Peter Bala, Jan 27 2020

From Peter Bala, Jan 10 2022: (Start)

exp(Sum_{n >= 1} a(n)*x^n/n) = 1 + 4*x + 17*x^2 + 76*x^3 + 354*x^4 + ... is the o.g.f. of A005572.

The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. (End)

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.

m-th binomial transforms of A000984: A126869 (m = -2), A002426 (m = -1 and m = -3 for signed version), A000984 (m = 0 and m = -4 for signed version), A026375 (m = 1 and m = -5 for signed version), A081671 (m = 2 and m = -6 for signed version), A098409 (m = 3 and m = -7 for signed version), A098410 (m = 4 and m = -8 for signed version), A104454 (m = 5 and m = -9 for signed version).

Sequence in context: A050146 A083879 A363184 * A244785 A260650 A006629

Adjacent sequences: A081668 A081669 A081670 * A081672 A081673 A081674

Keyword

nonn,easy

Author

Paul Barry, Mar 28 2003

Status

approved