A084605 G.f.: 1/(1-2x-15x^2)^(1/2); also, a(n) is the central coefficient of (1+x+4x^2)^n.

1, 1, 9, 25, 145, 561, 2841, 12489, 60705, 281185, 1353769, 6418809, 30917041, 148331665, 716698425, 3462260265, 16786700865, 81464917185, 396215601225, 1929237099225, 9408084660945, 45928695279345, 224476389327705

Offset

0,3

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 (or D) steps come in four colors. - N-E. Fahssi, Mar 30 2008

Ignoring initial term, equals the logarithmic derivative of A091147. - Paul D. Hanna, Dec 08 2018

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. - Peter Bala, Jan 10 2022

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1433 (terms 0..200 from Vincenzo Librandi)

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.

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

Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, Some matrix identities on colored Motzkin paths, Discrete Mathematics 340.12 (2017): 3081-3091.

Formula

E.g.f.: exp(x)*BesselI(0, 4*x). - Vladeta Jovovic, Aug 20 2003

a(n) is also the central coefficient of (4+x+x^2)^n; a(n) = Sum_{k=0..n} 3^(n-k) C(n,k) T(k,n), where T(k,n) is the triangle of trinomial coefficients = Coefficient of x^n of (1+x+x^2)^k : A027907. - N-E. Fahssi, Mar 30 2008

a(n) = (1/Pi)*integral(x=-2..2, (2*x+1)^n/sqrt((2-x)*(2+x))). - Peter Luschny, Sep 12 2011

D-finite with recurrence a(n+2) = ((2*n+3)*a(n+1) + 15*(n+1)*a(n))/(n+2); a(0)=a(1)=1 - Sergei N. Gladkovskii, Aug 01 2012

a(n) ~ 5^(n+1/2)/(2*sqrt(2*Pi*n)). - Vaclav Kotesovec, Oct 14 2012

a(n) = 2^n*GegenbauerC(n, -n, -1/4). - Peter Luschny, May 08 2016

a(n) = hypergeom([1/2 - n/2, -n/2], [1], 16). - Peter Luschny, Mar 18 2018

a(n) = Sum_{k=0..n} (-3)^(n-k) * 2^k * binomial(n,k)*binomial(2*k,k). - Paul D. Hanna, Dec 09 2018

a(n) = Sum_{k=0..n} 5^(n-k) * (-2)^k * binomial(n,k)*binomial(2*k,k). - Seiichi Manyama, May 01 2019

Maple

a := n -> simplify(2^n*GegenbauerC(n, -n, -1/4)):
seq(a(n), n=0..22); # Peter Luschny, May 08 2016

Mathematica

Table[n!*SeriesCoefficient[E^x*BesselI[0, 4*x], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 14 2012 *)
a[n_] := Hypergeometric2F1[1/2 - n/2, -n/2, 1, 16];
Table[a[n], {n, 0, 22}] (* Peter Luschny, Mar 18 2018 *)

Prog

(PARI) for(n=0, 30, t=polcoeff((1+x+4*x^2)^n, n, x); print1(t", "))
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = sum(k=0, n, (-3)^(n-k)*2^k*binomial(n, k)*binomial(2*k, k))}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Dec 09 2018

Crossrefs

Cf. A002426, A084600-A084604, A084606-A084615.

Cf. A322240 (a(n)^2), A091147.

Sequence in context: A227078 A146365 A146373 * A098773 A089998 A014728

Adjacent sequences: A084602 A084603 A084604 * A084606 A084607 A084608

Keyword

nonn

Author

Paul D. Hanna, Jun 01 2003

Status

approved