A098336 -id:A098336 - cp's OEIS frontend

A098332 Expansion of 1/sqrt(1 - 2x + 9x^2).

1, 1, -3, -11, 1, 81, 141, -363, -1791, -479, 13597, 29877, -54911, -353807, -223443, 2539989, 6806529, -8302527, -73999299, -73313931, 489731841, 1584548241, -1110170163, -15812965611, -21391839999, 94696016481

Offset: 0

Paul Barry, Sep 03 2004

Central coefficients of (1 + x - 2*x^2)^n.
Binomial transform of 1/sqrt(1+8*x^2), or (1,0,-4,0,24,0,...).
Binomial transform is A098336.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, ex. 7.56, p. 575.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
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.

Cf. A126869, A012000, A116091, A098341.

a := n -> hypergeom([1/2-n/2, -n/2], [1], -8);
seq(round(evalf(a(n),99)),n=0..30); # Peter Luschny, Sep 18 2014

Table[(-3)^n*LegendreP[n,-1/3],{n,0,20}] (* Vaclav Kotesovec, Jul 23 2013 *)
CoefficientList[Series[1/Sqrt[1 - 2*x + 9*x^2], {x,0,50}], x] (* G. C. Greubel, Feb 18 2017 *)

x='x+O('x^25); Vec(1/sqrt(1 - 2*x + 9*x^2)) \\ G. C. Greubel, Feb 18 2017

E.g.f.: exp(x)*BesselI(0, 2*sqrt(-2)*x);
a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * binomial(2*k,k) * (-2)^k.
a(n) = Sum_{k=0..floor(n/2)} binomial(n,k) * binomial(n-k,k) * (-2)^k.
a(n) = (-1)^n * Sum_{k=0..n} binomial(n,k)^2 * (-2)^k.
G.f.: A(x) = 1/(2*T(0)+3*x-1) where T(k) = 1 - 2*x/(1 + x/T(k+1)); (continued fraction, 2-step ). - Sergei N. Gladkovskii, Aug 23 2012
D-finite with recurrence: a(n+2) = ((2*n+3)*a(n+1))/(n+2) - (9*(n+1)*a(n))/(n+2) with a(0)=1, a(1)=1. (See Graham, Knuth, and Patashnik). - Alexander R. Povolotsky, Aug 23 2012
a(n) = hypergeom([1/2-n/2, -n/2], [1], -8). - Peter Luschny, Sep 18 2014
a(n) = (3/2)*(9/2)^n*Sum_{k >= 0} (-1/2)^k*binomial(n+k,k)^2. - Peter Bala, Mar 02 2017

A116093 Expansion of 1/sqrt(1+4x+12x^2).

Original entry on oeis.org

1, -2, 0, 16, -56, 48, 384, -1920, 3168, 8512, -66560, 161280, 113920, -2224640, 7311360, -3354624, -69253632, 306754560, -408059904, -1898029056, 12054196224, -25377005568, -38874316800, 443400781824, -1289598418944, -52751204352, 15086928789504, -58620595404800

Offset: 0

Paul Barry, Feb 04 2006

Apart from signs identical this is to A098336. - Joerg Arndt, Jun 30 2013

Fourth binomial transform of the expansion of 1/sqrt(1-4*x+12*x^2), A098336.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.

Column 2 of A307819.

Cf. A098336.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt(1+4*x+12*x^2) )); // G. C. Greubel, May 10 2019

CoefficientList[Series[1/Sqrt[1+4x+12x^2],{x,0,30}],x] (* Harvey P. Dale, Oct 15 2014 *)

my(x='x+O('x^30)); Vec(1/sqrt(1+4*x+12*x^2)) \\ G. C. Greubel, May 10 2019

(1/sqrt(1+4*x+12*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 10 2019

E.g.f.: exp(-2*x)*Bessel_I(0,2*sqrt(-2)*x).
a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*binomial(n-k,k)(-2)^(n-k).
D-finite with recurrence: n*a(n) +2*(2*n-1)*a(n-1) +12*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 07 2012
G.f.:  G(0), where G(k)= 1 - 2*x*(1+3*x)*(4*k+1)/( 2*k+1 - x*(1+3*x)*(2*k+1)*(4*k+3)/(x*(1+3*x)*(4*k+3) - (k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jun 30 2013

A098339 Expansion of 1/sqrt(1 - 6x + 17x^2).

Original entry on oeis.org

1, 3, 5, -9, -111, -477, -1051, 1095, 21793, 106947, 276165, -71145, -4712655, -26071965, -76452315, -29748249, 1045547073, 6564746115, 21507513221, 19922192439, -230801512751, -1674387214173, -6072718662555

Offset: 0

Paul Barry, Sep 03 2004

Binomial transform of A098336. Second binomial transform of A098332.

Central coefficients of (1 + 3x - 2x^2)^n.

Michael De Vlieger, Table of n, a(n) for n = 0..1628
Hacène Belbachir, Abdelghani Mehdaoui, 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.

CoefficientList[Series[1/Sqrt[1-6x+17x^2],{x,0,30}],x] (* Harvey P. Dale, Jun 19 2013 *)

E.g.f.: exp(3x)*BesselI(0, 2*sqrt(-2)*x).

D-finite with recurrence: n*a(n) + 3*(1-2*n)*a(n-1) + 17*(n-1)*a(n-2) = 0. - R. J. Mathar, Nov 09 2012

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A098332 Expansion of 1/sqrt(1 - 2x + 9x^2).

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A098332 Expansion of 1/sqrt(1 - 2*x + 9*x^2).

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A098332 Expansion of 1/sqrt(1 - 2x + 9x^2).

A116093 Expansion of 1/sqrt(1+4x+12x^2).