A098331 -id:A098331 - cp's OEIS frontend

A098335 Expansion of 1/sqrt(1-4x+8x^2).

1, 2, 2, -4, -26, -68, -76, 184, 1222, 3308, 3772, -9656, -64676, -177448, -203992, 536176, 3607622, 9968972, 11510636, -30723416, -207302156, -575382392, -666187432, 1796105744, 12142184476, 33803271032

Offset: 0

Paul Barry, Sep 03 2004

Central coefficients of (1+2x-x^2)^n. Binomial transform of A098331.

Diagonal of rational function 1/(1 - (x^2 + 2*x*y - y^2)). - Gheorghe Coserea, Aug 04 2018

G. C. Greubel, Table of n, a(n) for n = 0..1000
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

a[n_] := 2^n*Hypergeometric2F1[1/2 - n/2, -n/2, 1, -1]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Sep 05 2012, after Alexander R. Povolotsky *)
CoefficientList[Series[1/Sqrt[1 - 4*x + 8*x^2], {x,0,50}], x] (* G. C. Greubel, Feb 19 2017 *)

x='x+O('x^25); Vec(1/sqrt(1 - 4*x + 8*x^2)) \\ G. C. Greubel, Feb 19 2017

From Paul Barry, Sep 08 2004: (Start)
E.g.f. : exp(2*x)*BesselI(0, 2*I*x), I=sqrt(-1);
a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*binomial(n-k,k)*2^n*(-4)^(-k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*binomial(2(n-k),k)*(-2)^k. (End)
E.g.f. : exp(2*x)*BesselJ(0, 2*x). - Sergei N. Gladkovskii, Aug 22 2012
It appears that a(j+2) = (2*(2*j+1)*a(j+1))/(j+1)-(8*j*a(j))/(j+1), in case of re-indexing from 0 to 1. - Alexander R. Povolotsky, Aug 22 2012
D-finite with recurrence: a(n+2) = ((4*n+6)*a(n+1) - 8*(n+1)*a(n))/(n+2); a(0)=1,a(1)=2. - Sergei N. Gladkovskii, Aug 22 2012
a(n) = 2^n*2F_1(1/2-n/2, -n/2, 1, -1), where _2F_1(a,b;c;x) is the hypergeometric function. - _Alexander R. Povolotsky, Aug 22 2012

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

A307819 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 + 2kx + k(k+4)x^2).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, -1, 0, 1, -3, 0, 5, 0, 1, -4, 3, 16, -5, 0, 1, -5, 8, 27, -56, -11, 0, 1, -6, 15, 32, -189, 48, 41, 0, 1, -7, 24, 25, -416, 567, 384, -29, 0, 1, -8, 35, 0, -725, 2176, 189, -1920, -125, 0, 1, -9, 48, -49, -1080, 5625, -4864, -11259, 3168, 365, 0

Offset: 0

Seiichi Manyama, May 05 2019

			Square array begins:
   1,   1,     1,      1,      1,      1,      1, ...
   0,  -1,    -2,     -3,     -4,     -5,     -6, ...
   0,  -1,     0,      3,      8,     15,     24, ...
   0,   5,    16,     27,     32,     25,      0, ...
   0, -11,    48,    567,   2176,   5625,  11664, ...
   0,  41,   384,    189,  -4864, -24375, -74304, ...
   0, -29, -1920, -11259, -23552,   9375, 228096, ...

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

Columns k=0..3 give A000007, (-1)^n * A098331, A116093, (-1)^n * A098340.
Main diagonal gives A307911.
Cf. A307860, A307884, A307910.

A[n_, k_] := (-k)^n*Hypergeometric2F1[(1-n)/2, -n/2, 1, -4/k]; A[0, ] = 1; A[, 0] = 0; Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 07 2019 *)

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

A104505 Triangle, read by rows, equal to the right-hand side of the triangle A084610, with row n listing the coefficients of (1+x-x^2)^n: T(n,k) = [x^(n+k)] (1+x-x^2)^n, for n>=k>=0.

Original entry on oeis.org

1, 1, -1, -1, -2, 1, -5, 0, 3, -1, -5, 8, 2, -4, 1, 11, 15, -10, -5, 5, -1, 41, -6, -30, 10, 9, -6, 1, 29, -77, -14, 49, -7, -14, 7, -1, -125, -120, 112, 56, -70, 0, 20, -8, 1, -365, 117, 288, -126, -126, 90, 12, -27, 9, -1, -131, 770, 45, -540, 90, 228, -105, -30, 35, -10, 1, 1409, 946, -1265, -495, 858, 33, -363, 110, 55, -44

Offset: 0

Paul D. Hanna, Mar 11 2005

Matrix inverse is triangle A104509 and is related to Fibonacci numbers. Column 0 equals A098331, with g.f.: 1/sqrt(1-2*x+5*x^2). Column 1 equals A104506, with g.f.: ((1-x)/sqrt(1-2*x+5*x^2)-1)/(2*x). Row sums equal A104507. Absolute row sums equal A104508.

Array (1/sqrt(1-2x+5x^2), (1-x-sqrt(1-2x+5x^2))/(2x)), in Riordan array notation. Product of A120616 by A007318. - Paul Barry, Jun 17 2006

			Rows begin:
1;
1,-1;
-1,-2,1;
-5,0,3,-1;
-5,8,2,-4,1;
11,15,-10,-5,5,-1;
41,-6,-30,10,9,-6,1;
29,-77,-14,49,-7,-14,7,-1;
-125,-120,112,56,-70,0,20,-8,1;
-365,117,288,-126,-126,90,12,-27,9,-1;
-131,770,45,-540,90,228,-105,-30,35,-10,1; ...

P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.

Cf. A104509, A098331, A007440, A104506, A104507, A104508, A108044.

T[n_, k_] := Coefficient[(1 + x - x^2)^n, x, n + k];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 27 2019 *)

PARI
```
T(n,k)=if(n
				
```

T(n, 0) = A098331(n). T(n, 1) = n*A007440(n) (n>0).
Column k has e.g.f. exp(x)*Bessel_I(k,2*sqrt(-1)x)*(sqrt(-1))^k. - Paul Barry, Jun 17 2006
From Peter Bala, Jun 29 2015: (Start)
Matrix factorization in the Riordan group: ( 1/(1 - x), x/(1 - x) ) * ( 1/sqrt(1 + 4*x^2), (1 - sqrt(1 + 4*x^2))/(2*x) ) = A007318 * signed version of A108044.
Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = (1 - x - sqrt(1 - 2*x + 5*x^2))/(2*x) and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan).
T(n,k) = [x^(n-k)] f(x)^n with f(x) = x^2 + x - 1. In general the (n,k)th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)

A120617 Hankel transform of g.f. 1/sqrt(1+4x^2).

Original entry on oeis.org

1, -2, -4, 8, 16, -32, -64, 128, 256, -512, -1024, 2048, 4096, -8192, -16384, 32768, 65536, -131072, -262144, 524288, 1048576, -2097152, -4194304, 8388608, 16777216, -33554432, -67108864, 134217728, 268435456, -536870912, -1073741824, 2147483648, 4294967296, -8589934592

Offset: 0

Paul Barry, Jun 17 2006

Hankel transform of e.g.f. Bessel_I(0,2*sqrt(-1)*x) or (1,0,-2,0,6,0,-20,...).
Hankel transform of Sum{k=0..n} (-1)^(n-k)*C(n,k)^2.
Hankel transform of A098331.
Hankel transform of A082590. - Paul Barry, Apr 26 2009

Index entries for linear recurrences with constant coefficients, signature (0,-4).

Cf. A082590, A098331.

LinearRecurrence[{0,-4},{1,-2},40] (* or *) CoefficientList[ Series[ (1-2x)/(1+4x^2),{x,0,40}],x] (* Harvey P. Dale, Oct 12 2011 *)

G.f.: (1-2*x)/(1+4*x^2); a(n) = 2^n*(cos(Pi*(n+1)/2)+sin(Pi*(n+1)/2)).
a(0)=1, a(1)=-2, a(n)=-4*a(n-2). - Harvey P. Dale, Oct 12 2011
a(n) = ( 2*i^(n+1) )^n, where i=sqrt(-1). - Bruno Berselli, Oct 12 2011
E.g.f.: cos(2*x) - sin(2*x). - Arkadiusz Wesolowski, Aug 31 2012

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).

A329073 a(n) = (1/n)Sum_{k=0..n-1} (40k+13)(-1)^k50^(n-1-k)T_k(4,1)T_k(1,-1)^2, where T_k(b,c) denotes the coefficient of x^k in the expansion of (x^2+bx+c)^k.

Original entry on oeis.org

13, 219, 7858, 221525, 9253710, 375158958, 16882409364, 736344816813, 32964312771550, 1471835619627770, 66910145732699964, 3061043035494001682, 141458526138008430124, 6567714993530314856700, 306628434270114823521000, 14370411994543866356077725, 676259546148988495771751550

Offset: 1

Zhi-Wei Sun, Nov 03 2019

nonn

Conjecture 1: (i) a(n) is a positive integer for each n > 0; also, a(n) is odd if and only if n is a power of two. Moreover, we have the identity Sum_{k>=0} ((40k+13)/(-50)^k)*T_k(4,1)*T_k(1,-1)^2 = 55*sqrt(15)/(9*Pi).
(ii) Let p > 5 be a prime. Then Sum_{k=0..p-1} ((40k+13)/(-50)^k)*T_k(4,1)* T_k(1,-1)^2 == (p/3)*(12 + 5*Leg(3/p) + 22*Leg(p/15)) (mod p^2), where Leg(a/p) denotes the Legendre symbol. Also, for the sum S(p) = Sum_{k=0..p-1} T_k(4,1)* T_k(1,-1)^2/(-50)^k, if Leg(-5/p) = -1 then S(p) == 0 (mod p^2); if p == 1,9 (mod 20) and p = x^2 + 5*y^2 with x and y integers then S(p) == 4x^2-2p (mod p^2); if p == 3,7 (mod 20) and 2p = x^2 + 5*y^2 with x and y integers then S(p) == 2x^2-2p (mod p^2).
Conjecture 2: (i) For any n > 0, the number b(n):=(1/n)*Sum_{k=0..n-1} (40k+27)*(-6)^(n-1-k)*T_k(4,1)*T_k(1,-1)^2 is an integer. Moreover, b(n) is odd if and only if n is a power of two.
(ii) Let p > 3 be a prime. Then Sum_{k=0..p-1} ((40k+27)/(-6)^k)*T_k(4,1)* T_k(1,-1)^2 == (p/9)*(55*Leg(-5/p) + 198*Leg(3/p)-10) (mod p^2). Also, for the sum T(p) = Sum_{k=0..p-1} T_k(4,1)*T_k(1,-1)^2/(-6)^k, if Leg(-5/p) = -1 then T(p) == 0 (mod p^2); if p == 1,9 (mod 20) and p = x^2 + 5*y^2 with x and y integers then T(p) == Leg(p/3)*(4x^2-2p) (mod p^2); if p == 3,7 (mod 20) and 2p = x^2 + 5*y^2 with x and y integers then T(p) == Leg(p/3)(2p-2x^2) (mod p^2).

			a(1) = 13 since (40*0+13)*(-1)^0*50^(1-1-0)*T_0(4,1)*T_0(1,-1)^2/1 = 13/1 = 13.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100
Zhi-Wei Sun, On sums related to central binomial and trinomial coefficients, in: M. B. Nathanson (ed.), Combinatorial and Additive Number Theory: CANT 2011 and 2012, Springer Proc. in Math. & Stat., Vol. 101, Springer, New York, 2014, pp. 257-312. Also available from arXiv:1101.0600 [math.NT], 2011-2014.

Cf. A081671, A098331.

T[b_,c_,0]=1;T[b_,c_,1]=b;
T[b_,c_,n_]:=T[b,c,n]=(b(2n-1)T[b,c,n-1]-(b^2-4c)(n-1)T[b,c,n-2])/n;
a[n_]:=a[n]=Sum[(40k+13)(-1)^k*50^(n-1-k)*T[4,1,k]*T[1,-1,k]^2,{k,0,n-1}]/n;
Table[a[n],{n,1,20}]

A128058 Expansion of 1/((1-x)sqrt(1-2x+5x^2)).

Original entry on oeis.org

1, 2, 1, -4, -9, 2, 43, 72, -53, -418, -549, 860, 4161, 4006, -11619, -41104, -24989, 145046, 399571, 89796, -1723259, -3787914, 829841, 19739016, 34642741, -26909998, -219300587, -300591148, 467328447, 2369124842

Offset: 0

Paul Barry, Feb 13 2007

Partial sums of A098331. Binomial transform of A128057. Hankel transform is A128056.

CoefficientList[Series[1/((1-x)Sqrt[1-2x+5x^2]),{x,0,50}],x] (* Harvey P. Dale, Jun 23 2018 *)

a(n)=sum{k=0..n, sum{j=0..k, C(k,j)*C(k-j,j)*(-1)^j}}

D-finite with recurrence: n*a(n) +(-3*n+1)*a(n-1) +(7*n-6)*a(n-2) +5*(-n+1)*a(n-3)=0. - R. J. Mathar, Jan 23 2020

A374508 Expansion of 1/(1 - 2x + 5x^2)^(5/2).

Original entry on oeis.org

1, 5, 5, -35, -140, -84, 840, 2640, 495, -16445, -41041, 11375, 282100, 559300, -474300, -4399260, -6807225, 11062275, 63677075, 73363675, -208411280, -865816600, -665544100, 3475847700, 11129861925, 4130560161, -53332660395, -135538728395, 9634906640

Offset: 0

Seiichi Manyama, Jul 09 2024

Cf. A098331, A102840, A374509.

Cf. A245551.

a[n_]:= Pochhammer[n+1, 4]*Hypergeometric2F1[(1-n)/2, -n/2, 3, -4]/4!; Array[a,29,0] (* Stefano Spezia, Jul 10 2024 *)

a(n) = binomial(n+4, 2)/6*sum(k=0, n\2, (-1)^k*binomial(n+2, n-2*k)*binomial(2*k+2, k));

a(0) = 1, a(1) = 5; a(n) = ((2*n+3)*a(n-1) - 5*(n+3)*a(n-2))/n.
a(n) = (binomial(n+4,2)/6) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = Pochhammer(n+1, 4)*hypergeom([(1-n)/2, -n/2], [3], -4)/4!. - Stefano Spezia, Jul 10 2024
a(n) = (-1)^n * Sum_{k=0..n} 2^k * (5/2)^(n-k) * binomial(-5/2,k) * binomial(k,n-k). - Seiichi Manyama, Aug 23 2025

cp's OEIS Frontend

A098335 Expansion of 1/sqrt(1-4x+8x^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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A307819 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 + 2*k*x + k*(k+4)*x^2).

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A104505 Triangle, read by rows, equal to the right-hand side of the triangle A084610, with row n listing the coefficients of (1+x-x^2)^n: T(n,k) = [x^(n+k)] (1+x-x^2)^n, for n>=k>=0.

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Mathematica

PARI

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A120617 Hankel transform of g.f. 1/sqrt(1+4x^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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Mathematica

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A329073 a(n) = (1/n)*Sum_{k=0..n-1} (40k+13)*(-1)^k*50^(n-1-k)*T_k(4,1)*T_k(1,-1)^2, where T_k(b,c) denotes the coefficient of x^k in the expansion of (x^2+b*x+c)^k.

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A307819 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 + 2kx + k(k+4)x^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).

A329073 a(n) = (1/n)Sum_{k=0..n-1} (40k+13)(-1)^k50^(n-1-k)T_k(4,1)T_k(1,-1)^2, where T_k(b,c) denotes the coefficient of x^k in the expansion of (x^2+bx+c)^k.

A374508 Expansion of 1/(1 - 2x + 5x^2)^(5/2).