A084770 -id:A084770 - cp's OEIS frontend

A098453 Expansion of 1/sqrt(1 - 4x - 12x^2).

1, 2, 12, 56, 304, 1632, 9024, 50304, 283392, 1607168, 9167872, 52537344, 302239744, 1744412672, 10096263168, 58576306176, 340566147072, 1983765676032, 11574393962496, 67631502065664, 395710949228544, 2318088492023808, 13594307705438208, 79802741538422784, 468895276304695296

Offset: 0

Paul Barry, Sep 08 2004

Central coefficient of (1 + 2x + 4x^2)^n.
a(n) is the number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps can have 2 colors and the U steps can have 4 colors. - N-E. Fahssi, Mar 31 2008
a(n) is the number of 2 X n matrices with terms in {1,2,3}, same number of 1's in top and bottom rows, and no constant columns. For example, a(1)=2 counts the transposes of (2,3) and (3,2). The number of such matrices with k 1's in each row is binomial(n,2k) [choose columns containing 1's] * binomial(2k,k) [place 1's in these columns] * 2^n [place 2 or 3 in the topmost available spot in each column and the other of 2,3 in the other spot if not occupied by a 1]. - David Callan, Aug 25 2009

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
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. A000984, A084609, A084770.

seq(simplify((-2)^n*hypergeom([-n,1/2], [1], 4)),n=0..20); # Peter Luschny, Apr 26 2016
T := proc(n, k) option remember;
if n < 0 or k < 0 then 0
elif n = 0 then binomial(2*k, k)
else 2*(T(n-1, k+1) - T(n-1, k)) fi end:
a := n -> T(n, 0): seq(a(n), n=0..20); # Peter Luschny, Aug 23 2017

Table[SeriesCoefficient[1/Sqrt[1-4*x-12*x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 15 2012 *)

x='x+O('x^66); Vec(1/sqrt(1-4*x-12*x^2)) \\ Joerg Arndt, May 11 2013

G.f.: 1/sqrt((1+2*x)*(1-6*x)).
E.g.f.: exp(2*x)*BesselI(0, 4*x).
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(2*(n-k), n)*3^k.
D-finite with recurrence: a(n+2) = ((4*n+6)*a(n+1) + 12*(n+1)*a(n))/(n+2); a(0)=1, a(1)=2. - Sergei N. Gladkovskii, Jul 30 2012
a(n) ~ sqrt(3)*6^n/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 15 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
a(n) = (-2)^n*hypergeom([-n,1/2], [1], 4). - Peter Luschny, Apr 26 2016
a(n) = 2^n*GegenbauerC(n, -n, -1/2). - Peter Luschny, May 08 2016

A098269 a(n) = 2^n*P_n(4), 2^n times the Legendre polynomial of order n at 4.

Original entry on oeis.org

1, 8, 94, 1232, 16966, 240368, 3468844, 50712992, 748553926, 11131168688, 166498969924, 2502416381792, 37759888297756, 571681667171168, 8679980422677784, 132116085646644032, 2015249400937940806

Offset: 0

Paul Barry, Sep 01 2004

Central coefficients of (1+8x+15x^2)^n. 2^n*LegendreP(n,k) yields the central coefficients of (1+2kx+(k^2-1)x^2)^n, with g.f. 1/sqrt(1-4kx+4x^2).
16th binomial transform of 2^n*LegendreP(n,-4) = (-1)^n*A098269(n). - Paul Barry, Sep 03 2004
Diagonal of rational functions 1/(1 + x + 3*y + x*z - 2*x*y*z), 1/(1 - x + y + 3*x*z - 2*x*y*z), 1/(1 - x - x*y - 3*y*z - 2*x*y*z). - Gheorghe Coserea, Jul 03 2018

Vincenzo Librandi, 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.

Cf. A069835, A084773.

Table[SeriesCoefficient[1/Sqrt[1-16*x+4*x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)
a[n_] := 3^n*HypergeometricPFQ[{-n, -n}, {1}, 5/3]; Flatten[Table[a[n], {n,0,16}]] (* Detlef Meya, May 21 2024 *)

a(n)=pollegendre(n,4)<Charles R Greathouse IV, Oct 24 2011

{a(n)=sum(k=0, n, binomial(n, k)^2*3^k*5^(n-k))} \\ Paul D. Hanna, Sep 29 2012

G.f.: 1/sqrt(1-16x+4x^2).
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n, k)*binomial(2(n-k), n)*4^(n-2k).
E.g.f.: exp(8*x)*BesselI(0, 2*sqrt(15)*x), cf. A084770. - Vladeta Jovovic, Sep 01 2004
a(n) = Sum_{k=0..n} binomial(n,k)^2 * 3^k * 5^(n-k). - Paul D. Hanna, Sep 29 2012
D-finite with recurrence: n*a(n) = 8*(2*n-1)*a(n-1) - 4*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ sqrt(450+120*sqrt(15))*(8+2*sqrt(15))^n/(30*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012
a(n) = 3^n*hypergeom([-n, -n], [1], 5/3) = 5^n*hypergeom([-n, -n], [1], 3/5). - Detlef Meya, May 21 2024

A098443 Expansion of 1/sqrt(1-8x-4x^2).

Original entry on oeis.org

1, 4, 26, 184, 1366, 10424, 80996, 637424, 5064166, 40528984, 326251276, 2638751504, 21426682876, 174563719984, 1426219233416, 11681133293024, 95877105146246, 788433553532824, 6494463369141116, 53576199709855184

Offset: 0

Paul Barry, Sep 07 2004

Binomial transform of A098444. Second binomial transform of A084770. Third binomial transform of A098264.

			G.f. = 1 + 4*x + 26*x^2 + 184*x^3 + 1366*x^4 + 10424*x^5 + 80996*x^6 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
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.

Column k=2 of A386621.

Cf. A006139, A126869.

CoefficientList[Series[1/Sqrt[1 - 8*x - 4*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 15 2012, updated Mar 21 2024 *)

x='x+O('x^66); Vec(1/sqrt(1-8*x-4*x^2)) \\ Joerg Arndt, May 11 2013

E.g.f.: exp(4*x) * BesselI(0, 2*sqrt(5)*x).
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k) * binomial(2(n-k), n) * 2^(n-2k).
D-finite with recurrence: n*a(n) = 4*(2*n-1)*a(n-1) + 4*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 15 2012
a(n) ~ sqrt(50+20*sqrt(5))*(4+2*sqrt(5))^n/(10*sqrt(Pi*n)). Equivalently, a(n) ~ 2^(n-1/2) * phi^(3*n + 3/2) / (5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Oct 15 2012, updated Mar 21 2024
G.f.: 1/(1 - 2*x*(2+x)*Q(0)), where Q(k)= 1 + (4*k+1)*x*(2+x)/(k+1 - x*(2+x)*(2*k+2)*(4*k+3)/(2*x*(2+x)*(4*k+3) + (2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
G.f.: Q(0), where Q(k) = 1 + 2*x*(x+2)*(4*k+1)/( 2*k+1 - x*(x+2)*(2*k+1)*(4*k+3)/(x*(x+2)*(4*k+3) + (k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 16 2013
From Peter Bala, Mar 16 2024: (Start)
a(n) = (-2*i)^n * P(n, 2*i), where i = sqrt(-1) and P(n, x) denotes the n-th Legendre polynomial.
Sum_{n >= 1} (-1)^(n+1)*4^n/(n*a(n-1)*a(n)) = 2*arctan(1/2) = 2*A073000. (End)
From Seiichi Manyama, Aug 29 2025: (Start)
a(n) = Sum_{k=0..n} (2-i)^k * (2+i)^(n-k) * binomial(n,k)^2, where i is the imaginary unit.
a(n) = Sum_{k=0..floor(n/2)} 5^k * 4^(n-2*k) * binomial(n,2*k) * binomial(2*k,k).
a(n) = [x^n] (1+4*x+5*x^2)^n. (End)

A098444 Expansion of 1/sqrt(1-6x-11x^2).

Original entry on oeis.org

1, 3, 19, 117, 771, 5193, 35629, 247467, 1734931, 12250953, 87006249, 620818047, 4447016781, 31959556983, 230331965379, 1664043517557, 12047551338771, 87387014213433, 634918255153369, 4619923954541247, 33661450900419001

Offset: 0

Paul Barry, Sep 07 2004

Binomial transform of A084770. Second binomial transform of A098264. Binomial transform is A098443.

Coefficient of x^n in (1 + 3 x + 5 x^2)^n = number of paths from the origin to (n,0) with steps U=(1,1), H=(1,0) and D=(1,-1); U can have 5 colors and H can have 3 colors. - N-E. Fahssi, Jan 28 2008

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[SeriesCoefficient[1/Sqrt[1-6*x-11*x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 15 2012 *)

x='x+O('x^66); Vec(1/sqrt(1-6*x-11*x^2)) \\ Joerg Arndt, May 11 2013

E.g.f.: exp(3x)*BesselI(0, 2*sqrt(5)*x)
D-finite with recurrence: n*a(n) = 3*(2*n-1)*a(n-1) + 11*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 15 2012
a(n) ~ sqrt(50+15*sqrt(5))*(3+2*sqrt(5))^n/(10*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 15 2012

A098455 Expansion of 1/sqrt(1-4x-36x^2).

Original entry on oeis.org

1, 2, 24, 128, 1096, 7632, 60864, 461568, 3648096, 28551872, 226695424, 1799989248, 14380907776, 115126211072, 924791445504, 7444100947968, 60057602459136, 485388465196032, 3929580292706304, 31858982479331328, 258641677679947776, 2102242140708298752

Offset: 0

Paul Barry, Sep 08 2004

Define Q(n,x) = Sum_{k=0..floor(n/2)} binomial(n,k) * binomial(2(n-k),n) * x^(n-2k). Then a(n) = 3^n*Q(n,1/3). A084770(n) is 2^n*Q(n,1/2). Central coefficient of (1+2*x+10*x^2)^n.

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.

Cf. A387428.

Table[SeriesCoefficient[1/Sqrt[1-4*x-36*x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 15 2012 *)

x='x+O('x^66); Vec(1/sqrt(1-4*x-36*x^2)) \\ Joerg Arndt, May 11 2013

E.g.f.: exp(2*x) * BesselI(0, 2*sqrt(10)*x).
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(2(n-k), n)*9^k.
D-finite with recurrence: n*a(n) = 2*(2*n-1)*a(n-1) + 36*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 15 2012
a(n) ~ sqrt(50+5*sqrt(10))*(2+2*sqrt(10))^n/(10*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 15 2012
From Seiichi Manyama, Aug 30 2025: (Start)
a(n) = Sum_{k=0..n} (1-3*i)^k * (1+3*i)^(n-k) * binomial(n,k)^2, where i is the imaginary unit.
a(n) = Sum_{k=0..floor(n/2)} 10^k * 2^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). (End)

A387466 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 4x - (2k*x)^2).

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 1, 2, 8, 20, 1, 2, 14, 32, 70, 1, 2, 24, 68, 136, 252, 1, 2, 38, 128, 406, 592, 924, 1, 2, 56, 212, 1096, 2332, 2624, 3432, 1, 2, 78, 320, 2566, 7632, 13964, 11776, 12870, 1, 2, 104, 452, 5320, 20092, 60864, 83848, 53344, 48620

Offset: 0

Seiichi Manyama, Aug 29 2025

			Square array begins:
    1,    1,     1,     1,      1,      1,       1, ...
    2,    2,     2,     2,      2,      2,       2, ...
    6,    8,    14,    24,     38,     56,      78, ...
   20,   32,    68,   128,    212,    320,     452, ...
   70,  136,   406,  1096,   2566,   5320,   10006, ...
  252,  592,  2332,  7632,  20092,  44752,   88092, ...
  924, 2624, 13964, 60864, 210524, 607424, 1523724, ...

Columns k=0..4 give A000984, A006139, A084770, A098455, A098456.
Main diagonal gives A387467.
Cf. A386621.

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

A(n,k) = Sum_{j=0..n} (1-k*i)^j * (1+k*i)^(n-j) * binomial(n,j)^2, where i is the imaginary unit.
A(n,k) = Sum_{j=0..floor(n/2)} k^(2*j) * binomial(2*(n-j),n-j) * binomial(n-j,j).
n*A(n,k) = 2*(2*n-1)*A(n-1,k) + 4*k^2*(n-1)*A(n-2,k) for n > 1.
A(n,k) = Sum_{j=0..floor(n/2)} (k^2+1)^j * 2^(n-2*j) * binomial(n,2*j) * binomial(2*j,j).
A(n,k) = [x^n] (1 + 2*x + (k^2+1)*x^2)^n.
E.g.f. of column k: exp(2*x) * BesselI(0, 2*sqrt(k^2+1)*x).

A098456 Expansion of 1/sqrt(1-4x-64x^2).

Original entry on oeis.org

1, 2, 38, 212, 2566, 20092, 210524, 1884136, 18854854, 178415852, 1764019828, 17115907096, 169100140444, 1661540282456, 16458178007288, 162887627833552, 1618680238292294, 16095872154638156, 160435286115927044, 1600771362880092472, 15997473711080724916

Offset: 0

Paul Barry, Sep 08 2004

Define Q(n,x)=sum{k=0..floor(n/2), binomial(n,k)binomial(2(n-k),n)x^(n-2k)}. Then a(n)=4^n*Q(n,1/4). Central coefficients of (1+2*x+17*x^2)^n.

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.

Cf. A084770, A098455.

CoefficientList[Series[1/Sqrt[1-4x-64x^2],{x,0,30}],x] (* Harvey P. Dale, Dec 27 2011 *)

x='x+O('x^66); Vec(1/sqrt(1-4*x-64*x^2)) \\ Joerg Arndt, May 11 2013

E.g.f.: exp(2x) * BesselI(0, 2*sqrt(17)*x).
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(2(n-k), n)*16^k.
D-finite with recurrence: n*a(n) +2*(1-2*n)*a(n-1) +64*(1-n)*a(n-2)=0. - R. J. Mathar, Sep 26 2012
a(n) ~ sqrt(578+34*sqrt(17))*(2+2*sqrt(17))^n/(34*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 15 2012
From Seiichi Manyama, Aug 30 2025: (Start)
a(n) = Sum_{k=0..n} (1-4*i)^k * (1+4*i)^(n-k) * binomial(n,k)^2, where i is the imaginary unit.
a(n) = Sum_{k=0..floor(n/2)} 17^k * 2^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). (End)

cp's OEIS Frontend

A098453 Expansion of 1/sqrt(1 - 4*x - 12*x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A098269 a(n) = 2^n*P_n(4), 2^n times the Legendre polynomial of order n at 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A098443 Expansion of 1/sqrt(1-8*x-4*x^2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A098444 Expansion of 1/sqrt(1-6x-11x^2).

Views

Author

Keywords

Comments

Links

Programs

Mathematica

PARI

Formula

A098455 Expansion of 1/sqrt(1-4*x-36*x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A387466 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 4*x - (2*k*x)^2).

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A098456 Expansion of 1/sqrt(1-4*x-64*x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A098453 Expansion of 1/sqrt(1 - 4x - 12x^2).

A098443 Expansion of 1/sqrt(1-8x-4x^2).

A098455 Expansion of 1/sqrt(1-4x-36x^2).

A387466 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 4x - (2k*x)^2).

A098456 Expansion of 1/sqrt(1-4x-64x^2).