A084602 -id:A084602 - cp's OEIS frontend

A084601 Coefficients of 1/(1-2x-7x^2)^(1/2); also, a(n) is the central coefficient of (1+x+2*x^2)^n.

1, 1, 5, 13, 49, 161, 581, 2045, 7393, 26689, 97285, 355565, 1305745, 4808545, 17760965, 65753693, 243954113, 906758785, 3375949829, 12587460557, 46995614449, 175669746209, 657370655045, 2462383495357, 9232029156001, 34641994234561, 130090261856261

Offset: 0

Paul D. Hanna, Jun 01 2003

The Hankel transform (see A001906 for definition) of this sequence is A036442 : 1, 4, 32, 512, 16384, ... . - Philippe Deléham, Jul 03 2005
Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), U (or D) can have 2 colors. - N-E. Fahssi, Feb 05 2008
Diagonal of the rational function 1 / (1 - 2*x^2 - y^2 - x*y). - Ilya Gutkovskiy, Apr 22 2025

Seiichi Manyama, Table of n, a(n) for n = 0..1718 (terms 0..120 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.

Cf. A002426, A084600, A084602-A084615.

Cf. A007318, A027907, A036442.

a := n -> hypergeom([1/2 - n/2, -n/2], [1], 8):
seq(simplify(a(n)), n=0..26); # Peter Luschny, Mar 18 2018

CoefficientList[Series[1/Sqrt[1-2x-7x^2],{x,0,30}],x] (* Harvey P. Dale, Sep 18 2011 *)

a(n):=coeff(expand((1+x+2*x^2)^n),x,n);
makelist(a(n),n,0,26); /* Emanuele Munarini, Mar 02 2011 */

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

E.g.f.: exp(x)*BesselI(0, 2*sqrt(2)*x). - Vladeta Jovovic, Mar 21 2004
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*binomial(n, k)*2^k. - Paul Barry, Aug 26 2004
a(n) = Sum_{k=0..n} Trinomial(k, n)*Binomial(n, k), with Trinomial=A027907 and Binomial=A007318. - Ralf Stephan, Jan 28 2005
a(n) is also the central coefficient of (2+x+x^2)^n; a(n) = Sum_{k=0..n} 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, Feb 05 2008
a(n+2) = ((2*n+3)*a(n+1) + 7*(n+1)*a(n))/(n+2); a(0)=a(1)=1. - Sergei N. Gladkovskii, Aug 01 2012
G.f.: G(0), where G(k)= 1 + x*(2+7*x)*(4*k+1)/( 4*k+2 - x*(2+7*x)*(4*k+2)*(4*k+3)/(x*(2+7*x)*(4*k+3) + 4*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 06 2013
a(n) ~ sqrt(8+2*sqrt(2)) * (1+2*sqrt(2))^n / (4*sqrt(Pi*n)). - Vaclav Kotesovec, May 09 2014
a(n) = hypergeom([1/2 - n/2, -n/2], [1], 8). - Peter Luschny, Mar 18 2018

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

Original entry on oeis.org

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

Paul D. Hanna, Jun 01 2003

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
Diagonal of the rational function 1 / (1 - 3*x^2 - y^2 - x*y). - Ilya Gutkovskiy, Apr 22 2025

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.

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

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

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

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)

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A084601 Coefficients of 1/(1-2x-7x^2)^(1/2); also, a(n) is the central coefficient of (1+x+2*x^2)^n.

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

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cp's OEIS Frontend

A084601 Coefficients of 1/(1-2*x-7*x^2)^(1/2); also, a(n) is the central coefficient of (1+x+2*x^2)^n.

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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.

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A084601 Coefficients of 1/(1-2x-7x^2)^(1/2); also, a(n) is the central coefficient of (1+x+2*x^2)^n.

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