A084603 Coefficients of 1/sqrt(1 - 2*x - 11*x^2); also, a(n) is the central coefficient of (1 + x + 3*x^2)^n.

1, 1, 7, 19, 91, 331, 1441, 5797, 24739, 103411, 441397, 1876777, 8047909, 34533253, 148803487, 642228139, 2778852979, 12043194163, 52286516821, 227323871929, 989675651041, 4313712072241, 18822940658947, 82215245701519

Offset

0,3

Comments

5th binomial transform of 2^n*LegendreP(n,-2) (signed version of A069835). - Paul Barry, Sep 03 2004

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 three colors. - N-E. Fahssi, Feb 05 2008

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1542 (terms 0..200 from Vincenzo Librandi)

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_{k=0..floor(n/2)} binomial(n-k, k)*binomial(n, k)3^k. - Paul Barry, Aug 26 2004

Binomial transform is A084609. Hankel transform is 6^n*3^C(n,2). - Paul Barry, Sep 16 2006

a(n) = (1/Pi)*Integral_{x=1-2*sqrt(3)..1+2*sqrt(3)} x^n/sqrt(-x^2 + 2*x + 11). - Paul Barry, Sep 16 2006

From Paul Barry, Sep 16 2006: (Start)

a(n) = Sum_{k=0..floor(n/2)} C(n,2k)*C(2k,k)*3^k;

a(n) = Sum_{k=0..floor(n/2)} C(n,k)*C(n-k,k)*3^k. (End)

From N-E. Fahssi, Feb 05 2008: (Start)

a(n) is also the central coefficient of (3+x+x^2)^n;

a(n) = Sum_{k=0..n} 2^(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. (End)

D-finite with recurrence: a(n+2) = ( (2*n+3)*a(n+1) + 11*(n+1)*a(n) )/(n+2); a(0)=a(1)=1. - Sergei N. Gladkovskii, Aug 01 2012

a(n) ~ sqrt(18+3*sqrt(3))*(1+2*sqrt(3))^n/(6*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012

E.g.f.: exp(x)*BesselI(0, 2*sqrt(3)*x). - Paul D. Hanna, Nov 09 2014, after Vladeta Jovovic in A084601

From Peter Bala, Jan 07 2022: (Start)

O.g.f. A(x) = 1 + x*d/dx(log(B(x)), where B(x) = (1 - x - sqrt(1 - 2*x - 11*x^2))/(6*x^2) is the o.g.f. of A025237.

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)

Mathematica

Table[Sum[Binomial[n-k, k]*Binomial[n, k]*3^k, {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 14 2012 *)

Prog

(PARI) for(n=0, 30, t=polcoeff((1+x+3*x^2)^n, n, x); print1(t", "))

Crossrefs

Cf. A002426, A084600-A084602, A084604-A084615, A025237.

Sequence in context: A109880 A363668 A245489 * A088883 A262186 A346513

Adjacent sequences: A084600 A084601 A084602 * A084604 A084605 A084606

Keyword

nonn,easy

Author

Paul D. Hanna, Jun 01 2003

Status

approved