A098333 -id:A098333 - cp's OEIS frontend

A098331 Expansion of 1/sqrt(1 - 2x + 5x^2).

1, 1, -1, -5, -5, 11, 41, 29, -125, -365, -131, 1409, 3301, -155, -15625, -29485, 16115, 170035, 254525, -309775, -1813055, -2064655, 4617755, 18909175, 14903725, -61552739, -192390589, -81290561, 767919595, 1901796395, 28588201

Offset: 0

Paul Barry, Sep 03 2004

Central coefficients of (1+x-x^2)^n. Binomial transform of 1/sqrt(1+4x^2), or (1,0,-2,0,6,0,-20,...). Binomial transform is A098335. (-1)^nA098331(n) is the inverse binomial transform of (1,0,-2,0,6,0,-20,...).

Hankel transform is 2^n*(-1)^C(n+1,2). Hankel transform of 0,1,1,-1,-5,-5,... is F(n)*(-1)^C(n+2,2)*(2^n+0^n)/2. - Paul Barry, Jan 13 2009

Robert Israel, Table of n, a(n) for n = 0..2865
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
Eric Weisstein's World of Mathematics, Trinomial Coefficient

Cf. A098332, A098333, A098334.

A098331 := n -> hypergeom([-n/2, 1/2-n/2], [1], -4);
seq(round(evalf(A098331(n),99)),n=0..30); # Peter Luschny, Sep 18 2014
f:= gfun:-rectoproc({(5*n+5)*a(n)+(-3-2*n)*a(n+1)+(n+2)*a(n+2), a(0) = 1, a(1) = 1},a(n),remember):
map(f, [$0..50]); # Robert Israel, Jan 30 2018

a=b=1; Join[{a, b}, Table[c=((2n-1)b-5(n-1)a)/n; a=b; b=c; c, {n, 2, 30}]] (Noe)
CoefficientList[Series[1/Sqrt[1-2x+5x^2],{x,0,40}],x] (* Harvey P. Dale, Aug 17 2015 *)

my(x='x+O('x^99)); Vec(1/(1-2*x+5*x^2)^(1/2)) \\ Altug Alkan, Mar 18 2018

E.g.f.: exp(x)*BesselI(0, 2*i*x), i=sqrt(-1);
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*binomial(2k, k)*(-1)^k;
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k)*(-1)^k;
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(k, k/2)*cos(pi*k/2).
D-finite with recurrence: a(0)=a(1)=1, a(n) = ((2n-1)a(n-1)-5(n-1)a(n-2))/n. - T. D. Noe, Oct 19 2005
a(n) = hypergeom([-n/2, 1/2-n/2], [1], -4). - Peter Luschny, Sep 18 2014
a(n) ~ 5^(n/2 + 1/4) * cos((Pi*n - arctan(1/2) - n*arctan(4/3))/2) / sqrt(Pi*n). - Vaclav Kotesovec, Oct 31 2017
a(n) = (sqrt(5))^n*P(n,1/sqrt(5)), where P(n,x) is the Legendre polynomial of degree n. Note the general result (sqrt(4*m+1))^n*P(n, 1/sqrt(4*m+1)) = Sum_{k = 0..floor(n/2)} C(n,2*k)*C(2*k,k)(-m)^k due to Catalan. - Peter Bala, Mar 18 2018
G.f.: 1/(1 - x + 2*x^2/(1 - x + x^2/(1 - x + x^2/(1 - x + x^2/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Nov 19 2021
From Peter Bala, Feb 08 2022: (Start)
G.f.: A(x) = Sum_{n >= 0} (-1)^n*binomial(2*n,n)*x^(2*n)/(1 - x)^(2*n+1).
a(n)^2 = Sum_{k = 0..n} (-1)^k*5^(n-k)*binomial(2*k,k)*binomial(n,k)* binomial(n+k,k).
Sum_{n >= 0} (-1)^n*binomial(2*n,n)^2 * x^n/(1 - 5*x)^(2*n+1) = 1 + x + x^2 + 25*x^3 + 25*x^4 + 121*x^5 + ... is the g.f. of a(n)^2.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all prime p and positive integers n and k. (End)

Corrected by T. D. Noe, Oct 19 2005

A098334 Expansion of 1/sqrt(1-2x+17x^2).

Original entry on oeis.org

1, 1, -7, -23, 49, 401, 41, -5767, -11423, 65569, 299353, -441847, -5511791, -3665999, 79937417, 212712857, -861871423, -5076450239, 3966949049, 89482678313, 110424995569, -1233175514671, -4202194115863, 11830822055353, 91629438996001, -13485315511199

Offset: 0

Paul Barry, Sep 03 2004

Central coefficients of (1+x-4x^2)^n.
Binomial transform of 1/sqrt(1+16x^2), or (1,0,-8,0,96,0,-1280,...).
Binomial transform is A098337.

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

Cf. A098331, A098332, A098333.

a := n -> hypergeom([-n/2, 1/2-n/2], [1], -16);
seq(round(evalf(a(n),99)), n=0..28); # Peter Luschny, Sep 18 2014

CoefficientList[Series[1/Sqrt[1-2*x+17*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 09 2014 *)

x='x+O('x^99); Vec(1/(1-2*x+17*x^2)^(1/2)) \\ Altug Alkan, Mar 18 2018

E.g.f.: exp(x)*BesselI(0, 4*I*x), I=sqrt(-1);
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*binomial(2*k, k)*(-4)^k;
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k)*(-4)^k;
a(n) = Sum_{k=0..n} binomial(n, k)*binomial(k, floor(k/2))*cos(Pi*k/2)*2^k.
D-finite with recurrence: n*a(n) +(-2*n+1)*a(n-1) +17*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 24 2012
Lim sup n->oo |a(n)|^(1/n) = sqrt(17). - Vaclav Kotesovec, Feb 09 2014
a(n) = hypergeom([-n/2, 1/2-n/2], [1], -16). - Peter Luschny, Sep 18 2014
a(n) = (sqrt(17))^n*P(n,1/sqrt(17)), where P(n,x) is the Legendre polynomial of degree n. - Peter Bala, Mar 18 2018

A307860 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2x + (1+4k)*x^2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, -3, -5, 1, 1, 1, -5, -11, -5, 1, 1, 1, -7, -17, 1, 11, 1, 1, 1, -9, -23, 19, 81, 41, 1, 1, 1, -11, -29, 49, 211, 141, 29, 1, 1, 1, -13, -35, 91, 401, 181, -363, -125, 1, 1, 1, -15, -41, 145, 651, 41, -2015, -1791, -365, 1

Offset: 0

Seiichi Manyama, May 02 2019

			Square array begins:
   1,  1,    1,     1,     1,      1,      1, ...
   1,  1,    1,     1,     1,      1,      1, ...
   1, -1,   -3,    -5,    -7,     -9,    -11, ...
   1, -5,  -11,   -17,   -23,    -29,    -35, ...
   1, -5,    1,    19,    49,     91,    145, ...
   1, 11,   81,   211,   401,    651,    961, ...
   1, 41,  141,   181,    41,   -399,  -1259, ...
   1, 29, -363, -2015, -5767, -12459, -22931, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
T. D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Columns k=0..5 give A000012, A098331, A098332, A098333, A098334.
Main diagonal gives A307862.
Cf. A307819, A307855.

T[n_, k_] := Sum[If[k == j == 0, 1, (-k)^j] * Binomial[n, j] * Binomial[n-j, j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 13 2021 *)

A(n,k) is the coefficient of x^n in the expansion of (1 + x - k*x^2)^n.
A(n,k) = Sum_{j=0..floor(n/2)} (-k)^j * binomial(n,j) * binomial(n-j,j) = Sum_{j=0..floor(n/2)} (-k)^j * binomial(n,2*j) * binomial(2*j,j).
n * A(n,k) = (2*n-1) * A(n-1,k) - (1+4*k) * (n-1) * A(n-2,k).

A098340 Expansion of 1/sqrt(1 - 6x + 21x^2).

Original entry on oeis.org

1, 3, 3, -27, -189, -567, 189, 11259, 59859, 129033, -395847, -4730481, -19580211, -21264201, 258785523, 1917734373, 6051991059, -2659507911, -135544952151, -738957668337, -1618780564359, 5297724346923, 63513121347063, 266379249285873, 286776522444861, -3683959713627417

Offset: 0

Paul Barry, Sep 03 2004

Binomial transform of A012000. Second binomial transform of A098333.

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

Seiichi Manyama, 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. A012000, A098333.

a := n -> 3^n*hypergeom([1/2 - n/2, -n/2], [1], -4/3):
seq(simplify(a(n)), n=0..21); # Peter Luschny, Mar 19 2018

CoefficientList[Series[1/Sqrt[1-6x+21x^2],{x,0,30}],x] (* Harvey P. Dale, Oct 07 2012 *)

my(x = 'x + O('x^30)); Vec(1/sqrt(1-6*x+21*x^2)) \\ Jinyuan Wang, Sep 08 2019

E.g.f.: exp(3*x)*BesselI(0, 2*sqrt(-3)*x).
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k)*3^n*(-3)^(-k).
a(n) = 3^n*Sum{k=0..floor(n/2)} binomial(n, 2k)*binomial(2k, k)*(-3)^(-k).
D-finite with recurrence: n*a(n) + 3*(1-2*n)*a(n-1) + 21*(n-1)*a(n-2) = 0. - R. J. Mathar, Sep 26 2012
Lim sup n->infinity |a(n)|^(1/n) = sqrt(21). - Vaclav Kotesovec, Sep 29 2013
a(n) = 3^n*hypergeom([1/2 - n/2, -n/2], [1], -4/3). - Peter Luschny, Mar 19 2018

A127946 Hankel transform of central coefficients of (1+k*x-3x^2)^n, k arbitrary integer.

Original entry on oeis.org

1, -6, -108, 5832, 944784, -459165024, -669462604992, 2928229434235008, 38424226636031774976, -1512608105754026853705216, -178635992073339063368878599168, 63289660175631590117213474413627392, 67269440586795655766964092111705109663744

Offset: 0

Paul Barry, Feb 08 2007

Hankel transform of A098333. The Hankel transform of e.g.f. Bessel_I(0,2*sqrt(-3)x) and its k-th binomial transforms, are given by a(n). In general, the Hankel transform of e.g.f. Bessel_I(0,2*sqrt(m)x) and its binomial transforms is 2^n*m^C(n+1,2).

Let T_n denote the n X n matrix with T_n(i,j) = 3^min(i,j); then a(n) = ((-1)^floor((n+1)/2))*det(T_(n+1))/3. - Lechoslaw Ratajczak, May 16 2021

G. C. Greubel, Table of n, a(n) for n = 0..63

a(n) = A083667(n+1)/2.

[2^n*(-3)^Binomial(n+1,2): n in [0..30]]; // G. C. Greubel, May 03 2018

A127946[0] = 1; A127946[n_] := {1, -1, -1, 1}[[Mod[n, 4] + 1]] * 2^n * 3^(n(n + 1)/2); Table[A127946[n], {n, 0, 12}] (* Jean-François Alcover, Oct 04 2016 *)
Table[2^n*(-3)^Binomial[n+1,2], {n,0,30}] (* G. C. Greubel, May 03 2018 *)

a(n)=if((n-1)%4<2,-1,1)*2^n*3^(n*(n+1)/2) \\ Charles R Greathouse IV, Oct 04 2016

a(n) = (cos(Pi*n/2) - sin(Pi*n/2))*6^n*3^C(n,2) = 2^n*(-3)^C(n+1,2).

cp's OEIS Frontend

A098331 Expansion of 1/sqrt(1 - 2*x + 5*x^2).

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A098334 Expansion of 1/sqrt(1-2x+17x^2).

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A307860 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*x + (1+4*k)*x^2).

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A098340 Expansion of 1/sqrt(1 - 6x + 21x^2).

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A127946 Hankel transform of central coefficients of (1+k*x-3x^2)^n, k arbitrary integer.

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A098331 Expansion of 1/sqrt(1 - 2x + 5x^2).

A307860 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2x + (1+4k)*x^2).