A084605 -id:A084605 - cp's OEIS frontend

A322240 a(n) = A084605(n)^2, the square of the central coefficient in (1 + x + 4*x^2)^n.

1, 1, 81, 625, 21025, 314721, 8071281, 155975121, 3685097025, 79065004225, 1832690505361, 41201108978481, 955863424195681, 22002282841672225, 513656632397480625, 11987246142597870225, 281793325930991748225, 6636532731958908324225, 156986802654088221500625, 3721955785026092495600625, 88512056987508595608293025

Offset: 0

Paul D. Hanna, Dec 08 2018

nonn

The g.f. of A084605 is 1/sqrt(1 - 2*x - 15*x^2).

Ignoring initial term, a(n) equals the logarithmic derivative of A322241.

			G.f.: A(x) = 1 + x + 81*x^2 + 625*x^3 + 21025*x^4 + 314721*x^5 + 8071281*x^6 + 155975121*x^7 + 3685097025*x^8 + 79065004225*x^9 + ... + A084605(n)^2*x^n + ...
that is,
A(x) = 1 + x + 9^2*x^2 + 25^2*x^3 + 145^2*x^4 + 561^2*x^5 + 2841^2*x^6 + 12489^2*x^7 + 60705^2*x^8 + 281185^2*x^9 + ... + A084605(n)^2*x^n + ...

Seiichi Manyama, Table of n, a(n) for n = 0..717

Cf. A322241, A084605.

a[n_] := Sum[(-3)^(n-k) * 2^k * Binomial[n,k] * Binomial[2k,k], {k, 0, n}]^2; Array[a, 20, 0] (* Amiram Eldar, Dec 13 2018 *)

/* a(n) = A084605(n)^2 */
{a(n)=polcoeff(1/sqrt(1 - 2*x - 15*x^2 +x*O(x^n)), n)^2}
for(n=0, 20, print1(a(n), ", "))

/* Using AGM: */
{a(n)=polcoeff( 1 / 1 / agm(1 + 3*5*x, sqrt((1 - 3^2*x)*(1 - 5^2*x) +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

/* Square of binomial sum */
{a(n) = sum(k=0,n, (-3)^(n-k)*2^k*binomial(n,k)*binomial(2*k,k))^2}
for(n=0, 20, print1(a(n), ", "))

G.f.: 1 / AGM(1 + 15*x, sqrt((1 - 9*x)*(1 - 25*x)) ), where AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) is the arithmetic-geometric mean.
G.f.: 1 / AGM((1-3*x)*(1-5*x), (1+3*x)*(1+5*x)) = Sum_{n>=0} a(n) * x^(2*n).
a(n) = A084605(n)^2, where A084605(n) = Sum_{k=0..n} (-3)^(n-k) * 2^k * binomial(n,k)*binomial(2*k,k).
a(n) ~ 5^(2*n+1) / (8*Pi*n). - Vaclav Kotesovec, Sep 27 2019

A322241 G.f.: exp( Sum_{n>=1} A084605(n)^2 * x^n/n ), where A084605(n) is the central coefficient in (1 + x + 4*x^2)^n.

Original entry on oeis.org

1, 1, 41, 249, 6305, 77569, 1665321, 27724889, 574252417, 10958980929, 228679916905, 4671350051321, 99292476904609, 2107949882690241, 45658568907254505, 993562984208479193, 21876513296218002433, 484448162130512673665, 10812975015547281792937, 242647271141110287979513, 5477046865641884201456033

Offset: 0

Paul D. Hanna, Dec 08 2018

nonn

Compare to: exp( Sum_{n>=1} A084605(n) * x^n/n ) = (1-x - sqrt(1 - 2*x - 15*x^2))/(8*x^2), the g.f. of A091147.

Sequence A322240(n) = A084605(n)^2 has generating function 1 / AGM(1 + 15*x, sqrt((1 - 9*x)*(1 - 25*x)) ).

			G.f.: A(x) = 1 + x + 41*x^2 + 249*x^3 + 6305*x^4 + 77569*x^5 + 1665321*x^6 + 27724889*x^7 + 574252417*x^8 + 10958980929*x^9 + 228679916905*x^10 + ...
such that
log(A(x)) = x + 81*x^2/2 + 625*x^3/3 + 21025*x^4/4 + 314721*x^5/5 + 8071281*x^6/6 + 155975121*x^7/7 + 3685097025*x^8/8 + ... + A084605(n)^2 * x^n/n + ...
RELATED SERIES.
The g.f. of A084605 equals the series
1/sqrt(1 - 2*x - 15*x^2) = 1 + x + 9*x^2 + 25*x^3 + 145*x^4 + 561*x^5 + 2841*x^6 + 12489*x^7 + 60705*x^8 + 281185*x^9 + ... + A084605(n) * x^n/n + ...

Cf. A322240, A084605.

{a(n)=if(n==0, 1, polcoeff(exp(sum(m=1, n, polcoeff(1/sqrt(1 - 2*x - 15*x^2 +x*O(x^m)), m)^2 *x^m/m)+x*O(x^n)), n))}
for(n=0,30,print1(a(n),", "))

A091147 Expansion of (1-x-sqrt(1-2x-15x^2))/(8*x^2).

Original entry on oeis.org

1, 1, 5, 13, 57, 201, 861, 3445, 14897, 63313, 278389, 1223069, 5465065, 24513945, 111037005, 505298565, 2314343265, 10645982625, 49202944485, 228253816365, 1062783893145, 4964167491945, 23256852644925, 109249893866133, 514494575459217, 2428488338526961

Offset: 0

Paul Barry, Dec 22 2003

a(n) = A014433(n+1)/4.

Number of lattice paths in the first quadrant from (0,0) to (n,0) using only steps H=(1,0), U=(1,1) and D=(1,-1), where the U steps come in 4 colors (i.e. Motzkin paths with the up steps in 4 colors). Series reversion of x/(1+x+4*x^2). - Paul Barry, May 16 2005

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

Cf. A217275, A084605.

a := n -> simplify(2^n*GegenbauerC(n, -n-1, -1/4)/(n+1)):
seq(a(n), n=0..25); # Peter Luschny, May 08 2016

a[0] = 1; a[1] = 1; a[n_] := ((2*n + 1)*a[n - 1] - 15*(1 - n)*a[n - 2])/(n + 2); Table[a[n], {n, 0, 50}] (* T. D. Noe, Oct 02 2012 *)
CoefficientList[Series[(1 - x - Sqrt[1 - 2 x - 15 x^2]) / (8 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, May 10 2013 *)
a[n_] := Hypergeometric2F1[1/2 - n/2, -n/2, 2, 16];
Table[a[n], {n, 0, 25}] (* Peter Luschny, Mar 18 2018 *)

a(n):=2^n*coeff(expand((1+x/2+x^2)^(n+1)),x^n)/(n+1);
makelist(a(n),n,0,30); /* Emanuele Munarini, Apr 27 2012 */

my(x='x+O('x^66)); Vec((1-x-sqrt(1-2*x-15*x^2))/(8*x^2)) \\ Joerg Arndt, May 11 2013

G.f.: 2/(1-x+sqrt(1-2*x-15*x^2)).
G.f.: exp( Sum_{n>=1} A084605(n) * x^n/n ). - Paul D. Hanna, Dec 08 2018
a(n) = Sum_{k=0..n} binomial(n, k)*4^(k/2)*C(k/2)*(1+(-1)^k)/2, C(n)=A000108(n).
a(n) = Sum_{k=0..n} binomial(n, 2*k)*C(k)*4^k. - Paul Barry, May 16 2005
a(n) = Integral_{x=-2..2} (2*x+1)^n*sqrt((2-x)*(2+x))/(2*Pi) dx. - Peter Luschny, Sep 11 2011
a(n) = (2^n/(n+1))*[x^n] (1+x/2+x^2)^(n+1). - Emanuele Munarini, Apr 27 2012
E.g.f.: a(n) = n! * [x^n] exp(x)*BesselI(1, 4*x)/(2*x). - Peter Luschny, Aug 25 2012
D-finite with recurrence: (n+2)*a(n) -(2*n+1)*a(n-1) +15*(1-n)*a(n-2)=0. - R. J. Mathar, Sep 26 2012, [corrected by Vaclav Kotesovec, Sep 29 2012]
a(n) ~ 5/8*sqrt(10)/(n^(3/2)*sqrt(Pi))*5^n. - Vaclav Kotesovec, Sep 29 2012
a(n) = hypergeom([-n/2, (1-n)/2], [2], 16). - Peter Luschny, May 28 2014
a(n) = 2^n*GegenbauerC(n,-n-1, -1/4)/(n+1). - Peter Luschny, May 08 2016
G.f.: 1/(1 - x - 4*x^2/(1 - x - 4*x^2/(1 - x - 4*x^2/(1 - x - 4*x^2/(1 - ....))))), a continued fraction. - Ilya Gutkovskiy, May 26 2017

A098264 G.f.: 1/(1-2x-19x^2)^(1/2).

Original entry on oeis.org

1, 1, 11, 31, 211, 851, 4901, 22961, 124531, 623011, 3313201, 17086301, 90453661, 473616781, 2509264811, 13250049551, 70368250451, 373539254611, 1989045489281, 10597110956861, 56566637447401, 302196871378601, 1616570627763311, 8654955238504531, 46384344189261661

Offset: 0

Paul Barry, Aug 31 2004

Central coefficient of (1+x+5x^2)^n.

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U steps can have five colors. - N-E. Fahssi, Mar 31 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Cf. A084601, A084603, A084605.

Table[SeriesCoefficient[1/Sqrt[1-2*x-19*x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)

x='x+O('x^66); Vec(1/(1-2*x-19*x^2)^(1/2)) \\ Joerg Arndt, May 11 2013

E.g.f. : exp(x)*BesselI(0, 2*sqrt(5)x).
a(n) = sum{k=0..floor(n/2), binomial(n, k)*binomial(n-k, k)*5^k}.
a(n) = sum{k=0..floor(n/2), binomial(n, 2k)*binomial(2k, k)*5^k}.
n*a(n) +(1-2*n)*a(n-1) +19*(1-n)*a(n-2)=0. - R. J. Mathar, Nov 14 2011
a(n) ~ sqrt(50+5*sqrt(5))*(1+2*sqrt(5))^n/(10*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012

A307855 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, 3, 1, 1, 1, 5, 7, 1, 1, 1, 7, 13, 19, 1, 1, 1, 9, 19, 49, 51, 1, 1, 1, 11, 25, 91, 161, 141, 1, 1, 1, 13, 31, 145, 331, 581, 393, 1, 1, 1, 15, 37, 211, 561, 1441, 2045, 1107, 1, 1, 1, 17, 43, 289, 851, 2841, 5797, 7393, 3139, 1

Offset: 0

Seiichi Manyama, May 01 2019

			Square array begins:
   1,   1,    1,    1,     1,     1,     1, ...
   1,   1,    1,    1,     1,     1,     1, ...
   1,   3,    5,    7,     9,    11,    13, ...
   1,   7,   13,   19,    25,    31,    37, ...
   1,  19,   49,   91,   145,   211,   289, ...
   1,  51,  161,  331,   561,   851,  1201, ...
   1, 141,  581, 1441,  2841,  4901,  7741, ...
   1, 393, 2045, 5797, 12489, 22961, 38053, ...

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..6 give A000012, A002426, A084601, A084603, A084605, A098264, A098265.
Main diagonal gives A187018.
Cf. A110180, A292627, A307847, A307860, A307910.

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).
D-finite with recurrence: n * A(n,k) = (2*n-1) * A(n-1,k) - (1-4*k) * (n-1) * A(n-2,k).

A084606 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2x+2x^2)^n.

Original entry on oeis.org

1, 1, 2, 2, 1, 4, 8, 8, 4, 1, 6, 18, 32, 36, 24, 8, 1, 8, 32, 80, 136, 160, 128, 64, 16, 1, 10, 50, 160, 360, 592, 720, 640, 400, 160, 32, 1, 12, 72, 280, 780, 1632, 2624, 3264, 3120, 2240, 1152, 384, 64, 1, 14, 98, 448, 1484, 3752, 7448, 11776, 14896, 15008, 11872

Offset: 0

Paul D. Hanna, Jun 01 2003

			Rows:
{1},
{1,2,2},
{1,4,8,8,4},
{1,6,18,32,36,24,8},
{1,8,32,80,136,160,128,64,16},
{1,10,50,160,360,592,720,640,400,160,32},
{1,12,72,280,780,1632,2624,3264,3120,2240,1152,384,64},

Cf. A002426, A084600-A084605, A006139, A084608-A084615.

for(n=0,15, for(k=0,2*n,t=polcoeff((1+2*x+2*x^2)^n,k,x); print1(t",")); print(" "))

A307847 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^2)*x^2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 9, 7, 1, 1, 1, 19, 25, 19, 1, 1, 1, 33, 55, 145, 51, 1, 1, 1, 51, 97, 595, 561, 141, 1, 1, 1, 73, 151, 1729, 2611, 2841, 393, 1, 1, 1, 99, 217, 4051, 8001, 22141, 12489, 1107, 1, 1, 1, 129, 295, 8209, 19251, 105441, 119449, 60705, 3139, 1

Offset: 0

Seiichi Manyama, May 01 2019

			Square array begins:
   1,   1,     1,      1,      1,       1, ...
   1,   1,     1,      1,      1,       1, ...
   1,   3,     9,     19,     33,      51, ...
   1,   7,    25,     55,     97,     151, ...
   1,  19,   145,    595,   1729,    4051, ...
   1,  51,   561,   2611,   8001,   19251, ...
   1, 141,  2841,  22141, 105441,  369501, ...
   1, 393, 12489, 119449, 627873, 2319801, ...

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

Columns k=0..2 give A000012, A002426, A084605.
Main diagonal gives A307844.
Cf. A292627, A307855.

T[n_, k_] := Sum[If[k == j == 0, 1, k^(2*j)] * Binomial[n, 2*j] * Binomial[2*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 constant term in the expansion of (k/x + 1 + k*x)^n.
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..n} (1-2*k)^(n-j) * k^j * binomial(n,j) * binomial(2*j,j).
A(n,k) = Sum_{j=0..n} (1+2*k)^(n-j) * (-k)^j * binomial(n,j) * binomial(2*j,j).
A(n,k) = Sum_{j=0..floor(n/2)} k^(2*j) * binomial(n,2*j) * binomial(2*j,j).
n * A(n,k) = (2*n-1) * A(n-1,k) - (1-4*k^2) * (n-1) * A(n-2,k).

A110180 Triangle of generalized central trinomial coefficients.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 5, 1, 1, 1, 19, 13, 7, 1, 1, 1, 51, 49, 19, 9, 1, 1, 1, 141, 161, 91, 25, 11, 1, 1, 1, 393, 581, 331, 145, 31, 13, 1, 1, 1, 1107, 2045, 1441, 561, 211, 37, 15, 1, 1, 1, 3139, 7393, 5797, 2841, 851, 289, 43, 17, 1, 1

Offset: 0

Paul Barry, Jul 14 2005

Rows sums are A110181. Diagonal sums are A110182. Columns include central trinomial coefficients A002426, A084601, A084603, A084605, A098264. T(n,k) = central coefficient (1 + x + kx^2)^n.

			Rows begin
  1;
  1,  1;
  1,  1,  1;
  1,  3,  1,  1;
  1,  7,  5,  1,  1;
  1, 19, 13,  7,  1,  1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

T[n_, 0] := 1; T[n_, k_] := Sum[Binomial[n - k, j]*Binomial[n - k - j, j]*k^j, {j, 0, Floor[(n - k)/2]}]; Table[T[n, k], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Mar 05 2017 *)

Number triangle T(n, k) = Sum_{j=0..floor((n-k)/2)} C(n-k, j)*C(n-k-j, j)*k^j.

A322248 G.f.: 1/sqrt( (1 + 3x)(1 - 13*x) ).

Original entry on oeis.org

1, 5, 57, 605, 6961, 81525, 973545, 11765325, 143522145, 1763351525, 21789466777, 270509191485, 3371353189009, 42155188480085, 528587607974217, 6644129071092525, 83691484792766145, 1056178325362832325, 13351036742005533945, 169019946403985898525, 2142600388730167543281, 27193744661180635582005, 345520219114720175821737, 4394534009569783690837005, 55943630366450131877449761, 712778930909503993783945125

Offset: 0

Paul D. Hanna, Dec 10 2018

nonn

			G.f.: A(x) = 1 + 5*x + 57*x^2 + 605*x^3 + 6961*x^4 + 81525*x^5 + 973545*x^6 + 11765325*x^7 + 143522145*x^8 + 1763351525*x^9 + 21789466777*x^10 + ...
such that A(x)^2 = 1/(1 - 10*x - 39*x^2).
RELATED SERIES.
exp( Sum_{n>=1} a(n)*x^n/n ) = 1 + 5*x + 41*x^2 + 365*x^3 + 3537*x^4 + 35925*x^5 + 378105*x^6 + 4084925*x^7 + 45044129*x^8 + 504880805*x^9 + 5735247817*x^10 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..899
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 96.

Cf. A084605, A098409, A248168.

Cf. A322249.

CoefficientList[Series[1/Sqrt[((1+3x)(1-13x))],{x,0,30}],x] (* Harvey P. Dale, Jun 29 2021 *)

/* Using generating function: */
{a(n) = polcoeff( 1/sqrt((1 + 3*x)*(1 - 13*x) +x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))

/* Using binomial formula: */
{a(n) = sum(k=0,n, (-3)^(n-k)*4^k*binomial(n,k)*binomial(2*k,k))}
for(n=0,30,print1(a(n),", "))

/* Using binomial formula: */
{a(n) = sum(k=0,n, 13^(n-k)*(-4)^k*binomial(n,k)*binomial(2*k,k))}
for(n=0,30,print1(a(n),", "))

/* a(n) as a central coefficient */
{a(n) = polcoeff( (1 + 5*x + 16*x^2 +x*O(x^n))^n, n)}
for(n=0,30,print1(a(n),", "))

a(n) = Sum_{k=0..n} 13^(n-k) * (-4)^k * binomial(n,k)*binomial(2*k,k).
a(n) = Sum_{k=0..n} (-3)^(n-k) * 4^k * binomial(n,k)*binomial(2*k,k).
a(n) equals the (central) coefficient of x^n in (1 + 5*x + 16*x^2)^n.
a(n) ~ 13^(n + 1/2) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Dec 10 2018
D-finite with recurrence: n*a(n) = 5*(2*n-1)*a(n-1) + 39*(n-1)*a(n-2) for n > 1. - Seiichi Manyama, Apr 22 2019
a(n) = (1/4)^n * Sum_{k=0..n} (-3)^k * 13^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

A084604 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1 + x + 4x^2)^n.

Original entry on oeis.org

1, 1, 1, 4, 1, 2, 9, 8, 16, 1, 3, 15, 25, 60, 48, 64, 1, 4, 22, 52, 145, 208, 352, 256, 256, 1, 5, 30, 90, 285, 561, 1140, 1440, 1920, 1280, 1024, 1, 6, 39, 140, 495, 1206, 2841, 4824, 7920, 8960, 9984, 6144, 4096, 1, 7, 49, 203, 791, 2261, 6027, 12489, 24108, 36176

Offset: 0

Paul D. Hanna, Jun 01 2003

			Rows:
{1},
{1,1, 4},
{1,2, 9,  8, 16},
{1,3,15, 25, 60,  48,  64},
{1,4,22, 52,145, 208, 352, 256, 256},
{1,5,30, 90,285, 561,1140,1440,1920,1280,1024},
{1,6,39,140,495,1206,2841,4824,7920,8960,9984,6144,4096},

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A002426, A084600-A084603, A084605-A084615.

With[{eq=(1+x+4x^2)},Flatten[Table[CoefficientList[Expand[eq^n],x],{n,0,10}]]] (* Harvey P. Dale, May 19 2011 *)

for(n=0,10, for(k=0,2*n,t=polcoeff((1+x+4*x^2)^n,k,x); print1(t",")); print(" "))

cp's OEIS Frontend

A322240 a(n) = A084605(n)^2, the square of the central coefficient in (1 + x + 4*x^2)^n.

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A322241 G.f.: exp( Sum_{n>=1} A084605(n)^2 * x^n/n ), where A084605(n) is the central coefficient in (1 + x + 4*x^2)^n.

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A091147 Expansion of (1-x-sqrt(1-2*x-15*x^2))/(8*x^2).

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A098264 G.f.: 1/(1-2x-19x^2)^(1/2).

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A307855 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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Mathematica

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A084606 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2x+2x^2)^n.

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PARI

A307847 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^2)*x^2).

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Mathematica

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A110180 Triangle of generalized central trinomial coefficients.

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A091147 Expansion of (1-x-sqrt(1-2x-15x^2))/(8*x^2).

A307855 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).

A307847 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^2)*x^2).

A322248 G.f.: 1/sqrt( (1 + 3x)(1 - 13*x) ).