A098332 -id:A098332 - 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

A116091 Expansion of 1/sqrt(1+4x+16x^2).

Original entry on oeis.org

1, -2, -2, 28, -74, -92, 1324, -3656, -4826, 70228, -197372, -267896, 3921724, -11126936, -15347432, 225505648, -643622906, -897078476, 13214495764, -37869162392, -53170602284, 784672445368, -2255295815192, -3183829452272, 47051201187676, -135537088268792, -192142210448216

Offset: 0

Paul Barry, Feb 04 2006

Fourth binomial transform is expansion of 1/sqrt(1-4*x+16*x^2), A012000.

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.

Cf. A012000, A098332, A126869.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt(1+4*x+16*x^2) )); // G. C. Greubel, May 09 2019

a := n -> (-4)^n*hypergeom([-n,1+n],[1],1/4);
seq(simplify(a(n)), n=0..26); # Peter Luschny, May 09 2016

Table[4^n*LegendreP[n,-1/2],{n,0,30}] (* Vaclav Kotesovec, Jul 23 2013 *)
CoefficientList[Series[1/Sqrt[1+4x+16x^2],{x,0,30}],x] (* Harvey P. Dale, Jun 08 2015 *)

Vec(1/sqrt(1+4*x+16*x^2+O(x^30))) \\ M. F. Hasler, Aug 25 2012

(1/sqrt(1+4*x+16*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 09 2019

E.g.f.: exp(-2*x)*Bessel_I(0, 2*sqrt(-3)*x).
a(n) = Sum_{k=0..n} C(n,k)^2*(-3)^k.
O.g.f.: P(-1/2,4*x) with the o.g.f. P(x,z):=1/sqrt(1-2*x*z+z^2) for the Legendre polynomials. Wolfdieter Lang, Mar 10 2011.
G.f. A(x) = 1/(2*T(0)-4*x-1) where T(k)= 1 + 3*x/(1 - x/T(k+1)); (continued fraction, 2-step ). - Sergei N. Gladkovskii, Aug 23 2012
D-finite with recurrence: a(n+2) = -(16*(n+1)*a(n))/(n+2) - (2*(2*n+3)*a(n+1))/(n+2) with a(0)=1, a(1)=-2. - Alexander R. Povolotsky, Aug 23 2012
a(n) = (-4)^n*hypergeom([-n, 1+n], [1], 1/4). - Peter Luschny, May 09 2016
From Peter Bala, Nov 30 2021: (Start)
a(n) = (-4)^n^P(n,1/2), where P(n,x) is the n-th Legendre polynomial.
a(n) = (4/3)*(16^n)*Sum_{k >= n} C(k,n)^2*(-1/3)^k.
a(n) = (-3)^n*hypergeom([-n, -n], [1], -1/3).
a(n) = (4/3)*(-16/3)^n*hypergeom([n+1, n+1], [1], -1/3).
a(n) = [x^n] ((1 + x)*(3 - x))^n. (End)

A307884 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 + 2(k-1)x + ((k+1)*x)^2).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, -2, 1, 1, -2, -3, 0, 1, 1, -3, -2, 11, 6, 1, 1, -4, 1, 28, 1, 0, 1, 1, -5, 6, 45, -74, -81, -20, 1, 1, -6, 13, 56, -255, -92, 141, 0, 1, 1, -7, 22, 55, -554, 477, 1324, 363, 70, 1, 1, -8, 33, 36, -959, 2376, 2689, -3656, -1791, 0, 1

Offset: 0

Seiichi Manyama, May 02 2019

Column k is the diagonal of the rational function 1 / ((1-x)*(1-y) + k*x*y). - Seiichi Manyama, Jul 11 2020

More generally, column k is the diagonal of the rational function r / ((1-r*x)*(1-r*y) + r-1 + (k-r+1)*r*x*y) for any nonzero real number r. - Seiichi Manyama, Jul 22 2020

			Square array begins:
  1,   1,   1,    1,    1,    1,      1, ...
  1,   0,  -1,   -2,   -3,   -4,     -5, ...
  1,  -2,  -3,   -2,    1,    6,     13, ...
  1,   0,  11,   28,   45,   56,     55, ...
  1,   6,   1,  -74, -255, -554,   -959, ...
  1,   0, -81,  -92,  477, 2376,   6475, ...
  1, -20, 141, 1324, 2689, -804, -20195, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=2..4 give (-1)^n * A098332, A116091, (-1)^n * A098341.
Main diagonal gives A307885.
T(n,n-1) gives A335310.
Cf. A307883, A336179.

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

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

A012000 Expansion of 1/sqrt(1 - 4x + 16x^2).

Original entry on oeis.org

1, 2, -2, -28, -74, 92, 1324, 3656, -4826, -70228, -197372, 267896, 3921724, 11126936, -15347432, -225505648, -643622906, 897078476, 13214495764, 37869162392, -53170602284, -784672445368, -2255295815192, 3183829452272, 47051201187676, 135537088268792, -192142210448216

Offset: 0

Wouter Meeussen

sign

			G.f. = 1 + 2*x - 2*x^2 - 28*x^3 - 74*x^4 + 92*x^5 + 1324*x^6 + 3656*x^7 + ...

T. D. Noe, Table of n, a(n) for n = 0..200
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. A116091, A098332, A126869.

a := n -> 4^n*hypergeom([-n,1+n],[1],1/4);
seq(simplify(a(n)),n=0..26); # Peter Luschny, May 09 2016

Table[ 2^(2n) LegendreP[ n, 1/2 ], {n, 12} ]

{a(n) = 2^(2*n) * subst( pollegendre(n), x, 1/2)} /* Michael Somos, Dec 03 2001 */

a(n) = polcoeff(((1 - x)*(1 + 3*x))^n, n); \\ Michel Marcus, Aug 16 2015

G.f.: 1/sqrt(1 - 4*x + 16*x^2). - Vladeta Jovovic, May 13 2003
Scaled Legendre polynomials evaluated at 1/2: 2^(2n)P(n, 1/2). - Michael Somos, Dec 03 2001
a(n) = (-1)^n*Sum_{k=0..n} C(n,k)^2*(-3)^k. - Benoit Cloitre, Oct 25 2003
a(n) = Sum_{k=0..floor(n/2)} C(n,k)*C(2(n-k),n)(-4)^k. - Paul Barry, Sep 08 2004
D-finite with recurrence: n*a(n) + 2*(1-2*n)*a(n-1) + 16*(n-1)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011
G.f.: G(0), where G(k) = 1 + x*(1-x)*(4*k+1)/( 4*k+2 - x*(1-x)*(4*k+2)*(4*k+3)/(x*(1-x)*(4*k+3) + 4*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 06 2013
a(n) ~ 2^(2*n+1)*sin(Pi/12*(4*n+5))/(3^(1/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Jul 30 2013
a(n) = [x^n] ((1 - x)*(1 + 3*x))^n. - Peter Bala, Aug 16 2015
a(n) = 4^n*hypergeom([-n,1+n], [1], 1/4). - Peter Luschny, May 09 2016
From Peter Bala, Nov 28 2021: (Start)
a(n) = 4^(2*n+1)*Sum_{k >= n} (-1)^(n+k)*C(k,n)^2*(1/3)^(k+1).
a(n) = (-1)^n*hypergeom([-n,-n], [1], -3).
a(n) = (4/3)*(16/3)^n*hypergeom([n+1,n+1], [1], -1/3). (End)

More terms from Vladeta Jovovic, May 13 2003

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

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).

A098336 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

Offset: 0

Paul Barry, Sep 03 2004

Central coefficients of (1 + 2*x - 2*x^2)^n.
Binomial transform of A098332.
Diagonal of rational function 1/(1 - (2*x^2 + 2*x*y - y^2)). - Gheorghe Coserea, Aug 04 2018

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

A098336 := n -> 2^n*hypergeom([-n/2, 1/2-n/2], [1], -2);
seq(round(evalf(A098336(n),99)),n=0..30); # Peter Luschny, Sep 18 2014

CoefficientList[Series[1/Sqrt[1-4*x+12*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 09 2013 *)

Vec(1/sqrt(1-4*x+12*x^2) + O(x^50)) \\ G. C. Greubel, Jan 30 2017

E.g.f.: exp(2*x)*BesselI(0, 2*sqrt(-2)*x).
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k)* 2^n* (-2)^(-k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(2(n-k), k)*(-3)^k. - Paul Barry, Sep 08 2004
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 24 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
Lim sup_{n->infinity} |a(n)|/((2*sqrt(3))^n/sqrt(Pi*n)) = 6^(1/4). - Vaclav Kotesovec, Oct 09 2013
a(n) = 2^n*hypergeom([-n/2, 1/2-n/2], [1], -2). - Peter Luschny, Sep 18 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)

A127945 Hankel transform of central coefficients of (1+k*x-2x^2)^n, k arbitrary integer.

Original entry on oeis.org

1, -4, -32, 512, 16384, -1048576, -134217728, 34359738368, 17592186044416, -18014398509481984, -36893488147419103232, 151115727451828646838272, 1237940039285380274899124224, -20282409603651670423947251286016

Offset: 0

Paul Barry, Feb 08 2007

Hankel transform of A098332. The Hankel transform of e.g.f. Bessel_I(0,2*sqrt(-2)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).

Unsigned version is A036442. - Philippe Deléham, Dec 11 2008

G. C. Greubel, Table of n, a(n) for n = 0..79

Cf. A036442, A098332.

[2^n*(-2)^Binomial(n+1,2): n in [0..25]]; // G. C. Greubel, May 01 2018

Table[2^n*(-2)^Binomial[n+1,2], {n, 0, 25}] (* G. C. Greubel, May 01 2018 *)

for(n=0,25, print1(2^n*(-2)^binomial(n+1,2), ", ")) \\ G. C. Greubel, May 01 2018

a(n) = (cos(Pi*n/2) - sin(Pi*n/2))*4^n*2^C(n,2).

a(n) = 2^n*(-2)^C(n+1,2).

cp's OEIS Frontend

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

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A116091 Expansion of 1/sqrt(1+4*x+16*x^2).

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A307884 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 + 2*(k-1)*x + ((k+1)*x)^2).

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A012000 Expansion of 1/sqrt(1 - 4*x + 16*x^2).

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A098333 Expansion of 1/sqrt(1 - 2x + 13x^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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A098331 Expansion of 1/sqrt(1 - 2x + 5x^2).

A116091 Expansion of 1/sqrt(1+4x+16x^2).

A307884 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 + 2(k-1)x + ((k+1)*x)^2).

A012000 Expansion of 1/sqrt(1 - 4x + 16x^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).

A098336 Expansion of 1/sqrt(1 - 4x + 12x^2).

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