A098334 -id:A098334 - 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

A098333 Expansion of 1/sqrt(1 - 2x + 13x^2).

Original entry on oeis.org

1, 1, -5, -17, 19, 211, 181, -2015, -5837, 12259, 91585, 29965, -1033955, -2347955, 7953115, 43864543, -11941037, -559875245, -942036911, 5060812717, 21502740649, -20676139991, -307241918945, -344022187613

Offset: 0

Paul Barry, Sep 03 2004

Central coefficients of (1 + x - 3x^2)^n.
Binomial transform of 1/sqrt(1+12x^2), or (1,0,-6,0,54,0,-540,...).
Binomial transform is A012000.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
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.

Cf. A098331, A098332, A098334.

a := n -> hypergeom([1/2 - n/2, -n/2], [1], -12):
seq(simplify(a(n)), n=0..23); # Peter Luschny, Mar 19 2018

CoefficientList[Series[1/Sqrt[1-2*x+13*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 09 2014 *)

x='x+O('x^99); Vec(1/(1-2*x+13*x^2)^(1/2)) \\ Altug Alkan, Mar 18 2018

E.g.f.: exp(x)*BesselI(0, 2*sqrt(-3)x);
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*binomial(2k, k)(-3)^k;
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k)(-3)^k.
D-finite with recurrence: n*a(n) + (-2*n+1)*a(n-1) + 13*(n-1)*a(n-2) = 0. - R. J. Mathar, Nov 24 2012
Lim sup n->infinity |a(n)|^(1/n) = sqrt(13). - Vaclav Kotesovec, Feb 09 2014
a(n) = (sqrt(13))^n*P(n,1/sqrt(13)), where P(n,x) is the Legendre polynomial of degree n. - Peter Bala, Mar 18 2018
a(n) = hypergeom([1/2 - n/2, -n/2], [1], -12). - Peter Luschny, Mar 19 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).

A098337 Expansion of 1/sqrt(1-4x+20x^2).

Original entry on oeis.org

1, 2, -4, -40, -80, 352, 2624, 3712, -32000, -186880, -134144, 2885632, 13520896, -1269760, -256000000, -966164480, 1056112640, 22286827520, 66722201600, -162411315200, -1901125959680, -4329895362560

Offset: 0

Paul Barry, Sep 03 2004

Central coefficients of (1+2x-4x^2)^n. Binomial transform of A098334.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
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.

Table[Sum[Binomial[n,k]*Binomial[2*(n-k), n]*(-5)^k,{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Feb 08 2014 *)
CoefficientList[Series[1/Sqrt[1-4x+20x^2],{x,0,30}],x] (* Harvey P. Dale, Jul 29 2015 *)

E.g.f.: exp(2x)BesselI(0, 4*I*x), I=sqrt(-1);
a(n) = sum{k=0..floor(n/2), binomial(n, k)binomial(n-k, k)2^n(-1)^k};
a(n) = sum{k=0..n, binomial(2k, k)binomial(k, n-k)(-5)^(n-k)}.
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)binomial(2(n-k), n)(-5)^k. - Paul Barry, Sep 08 2004.
D-finite with recurrence: n*a(n) +2*(-2*n+1)*a(n-1) +20*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 24 2012
Lim sup n->infinity |a(n)|^(1/n) = 2*sqrt(5). - Vaclav Kotesovec, Feb 08 2014

A098341 Expansion of 1/sqrt(1 - 6x + 25x^2).

Original entry on oeis.org

1, 3, 1, -45, -255, -477, 2689, 25203, 82945, -90045, -2379519, -11581677, -12063999, 197669475, 1423716225, 3645266355, -12180238335, -156702949245, -626511576575, 51239061075, 15179398450945, 87687927568035, 151934475887745

Offset: 0

Paul Barry, Sep 03 2004

Central coefficients of (1 + 3*x - 4*x^2)^n.
(-1)^n*a(n) is the sum of squares of coefficients of (1+2*i*x)^n where i=sqrt(-1) (see PARI code). - Joerg Arndt, Jul 06 2011
Binomial transform of A098337.
Second binomial transform of A098334.

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. A098332, A012000, A116091, A126869.

Table[(-5)^n*LegendreP[n,-3/5],{n,0,20}] (* Vaclav Kotesovec, Jul 23 2013 *)
CoefficientList[Series[1/Sqrt[1-6x+25x^2],{x,0,30}],x] (* Harvey P. Dale, Aug 22 2014 *)

a(n)={local(v=Vec((1+2*I*x)^n)); (-1)^n*sum(k=1,#v,v[k]^2);} /* Joerg Arndt, Jul 06 2011 */

a(n)={local(v=Vec((1+2*I*x)^n)); sum(k=1,#v, real(v[k])^2-imag(v[k])^2);} /* Joerg Arndt, Jul 06 2011 */

A098341 = lambda n: (-1)^n*hypergeometric([-n,-n], [1], -4)
[Integer(A098341(n).n(100)) for n in (0..22)] # Peter Luschny, Sep 23 2014

E.g.f.: exp(3*x)*BesselI(0, 4*I*x), I=sqrt(-1).
a(n) = (-1)^n*Sum_{k=0..n} binomial(n, k)^2*(-4)^k.
a(n) = (-1)^n*hypergeometric([-n,-n], [1], -4). - Peter Luschny, Sep 23 2014
D-finite with recurrence: n*a(n) +3*(-2*n+1)*a(n-1) +25*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 27 2014
From Peter Bala, Nov 28 2021: (Start)
a(n) = (5^n)*P(n,3/5), where P(n,x) is the n-th Legendre polynomial.
a(n) = [x^n] ((1 - x)*(1 + 4*x))^n.
a(n) = 5^(2*n+1)*Sum_{k >= n} (-1)^(n+k)*binomial(k,n)^2*(1/4)^(k+1).
a(n) = (5/4)*(25/4)^n*hypergeom([n+1, n+1], [1], -1/4). (End)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A098333 Expansion of 1/sqrt(1 - 2x + 13x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A098337 Expansion of 1/sqrt(1-4x+20x^2).

Views

Author

Keywords

Comments

Links

Programs

Mathematica

Formula

A098341 Expansion of 1/sqrt(1 - 6*x + 25*x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Sage

Formula

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).

A098341 Expansion of 1/sqrt(1 - 6x + 25x^2).